diff --git a/docs/html/user_guide.rst b/docs/html/user_guide.rst
index 61e0cdf0769..c36ac520528 100644
--- a/docs/html/user_guide.rst
+++ b/docs/html/user_guide.rst
@@ -254,9 +254,11 @@ Constraints Files
 
 Constraints files are requirements files that only control which version of a
 requirement is installed, not whether it is installed or not. Their syntax and
-contents is nearly identical to :ref:`Requirements Files`. There is one key
-difference: Including a package in a constraints file does not trigger
-installation of the package.
+contents is a subset of :ref:`Requirements Files`, with several kinds of syntax
+not allowed: constraints must have a name, they cannot be editable, and they
+cannot specify extras. In terms of semantics, there is one key difference:
+Including a package in a constraints file does not trigger installation of the
+package.
 
 Use a constraints file like so:
 
diff --git a/news/8253.feature.rst b/news/8253.feature.rst
new file mode 100644
index 00000000000..196e4dd9613
--- /dev/null
+++ b/news/8253.feature.rst
@@ -0,0 +1 @@
+Add the ability for the new resolver to process URL constraints.
diff --git a/src/pip/_internal/models/link.py b/src/pip/_internal/models/link.py
index 6f9a3244383..86d0be4079d 100644
--- a/src/pip/_internal/models/link.py
+++ b/src/pip/_internal/models/link.py
@@ -240,3 +240,9 @@ def is_hash_allowed(self, hashes):
         assert self.hash is not None
 
         return hashes.is_hash_allowed(self.hash_name, hex_digest=self.hash)
+
+
+# TODO: Relax this comparison logic to ignore, for example, fragments.
+def links_equivalent(link1, link2):
+    # type: (Link, Link) -> bool
+    return link1 == link2
diff --git a/src/pip/_internal/req/constructors.py b/src/pip/_internal/req/constructors.py
index 810fc085988..b5e7e7c3e14 100644
--- a/src/pip/_internal/req/constructors.py
+++ b/src/pip/_internal/req/constructors.py
@@ -461,3 +461,19 @@ def install_req_from_parsed_requirement(
             user_supplied=user_supplied,
         )
     return req
+
+
+def install_req_from_link_and_ireq(link, ireq):
+    # type: (Link, InstallRequirement) -> InstallRequirement
+    return InstallRequirement(
+        req=ireq.req,
+        comes_from=ireq.comes_from,
+        editable=ireq.editable,
+        link=link,
+        markers=ireq.markers,
+        use_pep517=ireq.use_pep517,
+        isolated=ireq.isolated,
+        install_options=ireq.install_options,
+        global_options=ireq.global_options,
+        hash_options=ireq.hash_options,
+    )
diff --git a/src/pip/_internal/req/req_install.py b/src/pip/_internal/req/req_install.py
index 5eba2c2ae74..2c5434830d8 100644
--- a/src/pip/_internal/req/req_install.py
+++ b/src/pip/_internal/req/req_install.py
@@ -840,8 +840,8 @@ def check_invalid_constraint_type(req):
     problem = ""
     if not req.name:
         problem = "Unnamed requirements are not allowed as constraints"
-    elif req.link:
-        problem = "Links are not allowed as constraints"
+    elif req.editable:
+        problem = "Editable requirements are not allowed as constraints"
     elif req.extras:
         problem = "Constraints cannot have extras"
 
diff --git a/src/pip/_internal/resolution/resolvelib/base.py b/src/pip/_internal/resolution/resolvelib/base.py
index 81fee9b9e3e..d42bca8bfa4 100644
--- a/src/pip/_internal/resolution/resolvelib/base.py
+++ b/src/pip/_internal/resolution/resolvelib/base.py
@@ -4,7 +4,7 @@
 from pip._vendor.packaging.utils import canonicalize_name
 from pip._vendor.packaging.version import _BaseVersion
 
-from pip._internal.models.link import Link
+from pip._internal.models.link import Link, links_equivalent
 from pip._internal.req.req_install import InstallRequirement
 from pip._internal.utils.hashes import Hashes
 
@@ -20,24 +20,26 @@ def format_name(project, extras):
 
 
 class Constraint:
-    def __init__(self, specifier, hashes):
-        # type: (SpecifierSet, Hashes) -> None
+    def __init__(self, specifier, hashes, links):
+        # type: (SpecifierSet, Hashes, FrozenSet[Link]) -> None
         self.specifier = specifier
         self.hashes = hashes
+        self.links = links
 
     @classmethod
     def empty(cls):
         # type: () -> Constraint
-        return Constraint(SpecifierSet(), Hashes())
+        return Constraint(SpecifierSet(), Hashes(), frozenset())
 
     @classmethod
     def from_ireq(cls, ireq):
         # type: (InstallRequirement) -> Constraint
-        return Constraint(ireq.specifier, ireq.hashes(trust_internet=False))
+        links = frozenset([ireq.link]) if ireq.link else frozenset()
+        return Constraint(ireq.specifier, ireq.hashes(trust_internet=False), links)
 
     def __nonzero__(self):
         # type: () -> bool
-        return bool(self.specifier) or bool(self.hashes)
+        return bool(self.specifier) or bool(self.hashes) or bool(self.links)
 
     def __bool__(self):
         # type: () -> bool
@@ -49,10 +51,16 @@ def __and__(self, other):
             return NotImplemented
         specifier = self.specifier & other.specifier
         hashes = self.hashes & other.hashes(trust_internet=False)
-        return Constraint(specifier, hashes)
+        links = self.links
+        if other.link:
+            links = links.union([other.link])
+        return Constraint(specifier, hashes, links)
 
     def is_satisfied_by(self, candidate):
         # type: (Candidate) -> bool
+        # Reject if there are any mismatched URL constraints on this package.
+        if self.links and not all(_match_link(link, candidate) for link in self.links):
+            return False
         # We can safely always allow prereleases here since PackageFinder
         # already implements the prerelease logic, and would have filtered out
         # prerelease candidates if the user does not expect them.
@@ -94,6 +102,13 @@ def format_for_error(self):
         raise NotImplementedError("Subclass should override")
 
 
+def _match_link(link, candidate):
+    # type: (Link, Candidate) -> bool
+    if candidate.source_link:
+        return links_equivalent(link, candidate.source_link)
+    return False
+
+
 class Candidate:
     @property
     def project_name(self):
diff --git a/src/pip/_internal/resolution/resolvelib/candidates.py b/src/pip/_internal/resolution/resolvelib/candidates.py
index 035e118d022..ccd1129dfc9 100644
--- a/src/pip/_internal/resolution/resolvelib/candidates.py
+++ b/src/pip/_internal/resolution/resolvelib/candidates.py
@@ -8,7 +8,7 @@
 from pip._vendor.pkg_resources import Distribution
 
 from pip._internal.exceptions import HashError, MetadataInconsistent
-from pip._internal.models.link import Link
+from pip._internal.models.link import Link, links_equivalent
 from pip._internal.models.wheel import Wheel
 from pip._internal.req.constructors import (
     install_req_from_editable,
@@ -155,7 +155,7 @@ def __hash__(self):
     def __eq__(self, other):
         # type: (Any) -> bool
         if isinstance(other, self.__class__):
-            return self._link == other._link
+            return links_equivalent(self._link, other._link)
         return False
 
     @property
diff --git a/src/pip/_internal/resolution/resolvelib/factory.py b/src/pip/_internal/resolution/resolvelib/factory.py
index 3181d575336..f740734061d 100644
--- a/src/pip/_internal/resolution/resolvelib/factory.py
+++ b/src/pip/_internal/resolution/resolvelib/factory.py
@@ -32,6 +32,7 @@
 from pip._internal.models.link import Link
 from pip._internal.models.wheel import Wheel
 from pip._internal.operations.prepare import RequirementPreparer
+from pip._internal.req.constructors import install_req_from_link_and_ireq
 from pip._internal.req.req_install import InstallRequirement
 from pip._internal.resolution.base import InstallRequirementProvider
 from pip._internal.utils.compatibility_tags import get_supported
@@ -264,6 +265,46 @@ def find_candidates(
             if ireq is not None:
                 ireqs.append(ireq)
 
+        for link in constraint.links:
+            if not ireqs:
+                # If we hit this condition, then we cannot construct a candidate.
+                # However, if we hit this condition, then none of the requirements
+                # provided an ireq, so they must have provided an explicit candidate.
+                # In that case, either the candidate matches, in which case this loop
+                # doesn't need to do anything, or it doesn't, in which case there's
+                # nothing this loop can do to recover.
+                break
+            if link.is_wheel:
+                wheel = Wheel(link.filename)
+                # Check whether the provided wheel is compatible with the target
+                # platform.
+                if not wheel.supported(self._finder.target_python.get_tags()):
+                    # We are constrained to install a wheel that is incompatible with
+                    # the target architecture, so there are no valid candidates.
+                    # Return early, with no candidates.
+                    return ()
+            # Create a "fake" InstallRequirement that's basically a clone of
+            # what "should" be the template, but with original_link set to link.
+            # Using the given requirement is necessary for preserving hash
+            # requirements, but without the original_link, direct_url.json
+            # won't be created.
+            ireq = install_req_from_link_and_ireq(link, ireqs[0])
+            candidate = self._make_candidate_from_link(
+                link,
+                extras=frozenset(),
+                template=ireq,
+                name=canonicalize_name(ireq.name) if ireq.name else None,
+                version=None,
+            )
+            if candidate is None:
+                # _make_candidate_from_link returns None if the wheel fails to build.
+                # We are constrained to install this wheel, so there are no valid
+                # candidates.
+                # Return early, with no candidates.
+                return ()
+
+            explicit_candidates.add(candidate)
+
         # If none of the requirements want an explicit candidate, we can ask
         # the finder for candidates.
         if not explicit_candidates:
diff --git a/tests/functional/test_install_direct_url.py b/tests/functional/test_install_direct_url.py
index 23273774d16..e28a7e9b57e 100644
--- a/tests/functional/test_install_direct_url.py
+++ b/tests/functional/test_install_direct_url.py
@@ -1,5 +1,7 @@
 import re
 
+import pytest
+
 from pip._internal.models.direct_url import DIRECT_URL_METADATA_NAME, DirectUrl
 from tests.lib import _create_test_package, path_to_url
 
@@ -46,3 +48,24 @@ def test_install_archive_direct_url(script, data, with_wheel):
     assert req.startswith("simple @ file://")
     result = script.pip("install", req)
     assert _get_created_direct_url(result, "simple")
+
+
+@pytest.mark.network
+def test_install_vcs_constraint_direct_url(script, with_wheel):
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text(
+        "git+https://github.com/pypa/pip-test-package"
+        "@5547fa909e83df8bd743d3978d6667497983a4b7"
+        "#egg=pip-test-package"
+    )
+    result = script.pip("install", "pip-test-package", "-c", constraints_file)
+    assert _get_created_direct_url(result, "pip_test_package")
+
+
+def test_install_vcs_constraint_direct_file_url(script, with_wheel):
+    pkg_path = _create_test_package(script, name="testpkg")
+    url = path_to_url(pkg_path)
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text(f"git+{url}#egg=testpkg")
+    result = script.pip("install", "testpkg", "-c", constraints_file)
+    assert _get_created_direct_url(result, "testpkg")
diff --git a/tests/functional/test_install_reqs.py b/tests/functional/test_install_reqs.py
index d559e94be18..d8de6248b7d 100644
--- a/tests/functional/test_install_reqs.py
+++ b/tests/functional/test_install_reqs.py
@@ -405,7 +405,7 @@ def test_constraints_constrain_to_local_editable(
         expect_error=(resolver_variant == "2020-resolver"),
     )
     if resolver_variant == "2020-resolver":
-        assert 'Links are not allowed as constraints' in result.stderr
+        assert 'Editable requirements are not allowed as constraints' in result.stderr
     else:
         assert 'Running setup.py develop for singlemodule' in result.stdout
 
@@ -419,12 +419,8 @@ def test_constraints_constrain_to_local(script, data, resolver_variant):
         'install', '--no-index', '-f', data.find_links, '-c',
         script.scratch_path / 'constraints.txt', 'singlemodule',
         allow_stderr_warning=True,
-        expect_error=(resolver_variant == "2020-resolver"),
     )
-    if resolver_variant == "2020-resolver":
-        assert 'Links are not allowed as constraints' in result.stderr
-    else:
-        assert 'Running setup.py install for singlemodule' in result.stdout
+    assert 'Running setup.py install for singlemodule' in result.stdout
 
 
 def test_constrained_to_url_install_same_url(script, data, resolver_variant):
@@ -438,7 +434,11 @@ def test_constrained_to_url_install_same_url(script, data, resolver_variant):
         expect_error=(resolver_variant == "2020-resolver"),
     )
     if resolver_variant == "2020-resolver":
-        assert 'Links are not allowed as constraints' in result.stderr
+        assert 'Cannot install singlemodule 0.0.1' in result.stderr, str(result)
+        assert (
+            'because these package versions have conflicting dependencies.'
+            in result.stderr
+        ), str(result)
     else:
         assert ('Running setup.py install for singlemodule'
                 in result.stdout), str(result)
@@ -489,7 +489,7 @@ def test_install_with_extras_from_constraints(script, data, resolver_variant):
         expect_error=(resolver_variant == "2020-resolver"),
     )
     if resolver_variant == "2020-resolver":
-        assert 'Links are not allowed as constraints' in result.stderr
+        assert 'Constraints cannot have extras' in result.stderr
     else:
         result.did_create(script.site_packages / 'simple')
 
@@ -521,7 +521,7 @@ def test_install_with_extras_joined(script, data, resolver_variant):
         expect_error=(resolver_variant == "2020-resolver"),
     )
     if resolver_variant == "2020-resolver":
-        assert 'Links are not allowed as constraints' in result.stderr
+        assert 'Constraints cannot have extras' in result.stderr
     else:
         result.did_create(script.site_packages / 'simple')
         result.did_create(script.site_packages / 'singlemodule.py')
@@ -538,7 +538,7 @@ def test_install_with_extras_editable_joined(script, data, resolver_variant):
         expect_error=(resolver_variant == "2020-resolver"),
     )
     if resolver_variant == "2020-resolver":
-        assert 'Links are not allowed as constraints' in result.stderr
+        assert 'Editable requirements are not allowed as constraints' in result.stderr
     else:
         result.did_create(script.site_packages / 'simple')
         result.did_create(script.site_packages / 'singlemodule.py')
diff --git a/tests/functional/test_new_resolver.py b/tests/functional/test_new_resolver.py
index 16f9f4f4216..78a21a3c115 100644
--- a/tests/functional/test_new_resolver.py
+++ b/tests/functional/test_new_resolver.py
@@ -10,7 +10,9 @@
     create_basic_sdist_for_package,
     create_basic_wheel_for_package,
     create_test_package_with_setup,
+    path_to_url,
 )
+from tests.lib.path import Path
 from tests.lib.wheel import make_wheel
 
 
@@ -45,6 +47,24 @@ def assert_editable(script, *args):
         f"{args!r} not all found in {script.site_packages_path!r}"
 
 
+@pytest.fixture()
+def make_fake_wheel(script):
+
+    def _make_fake_wheel(name, version, wheel_tag):
+        wheel_house = script.scratch_path.joinpath("wheelhouse")
+        wheel_house.mkdir()
+        wheel_builder = make_wheel(
+            name=name,
+            version=version,
+            wheel_metadata_updates={"Tag": []},
+        )
+        wheel_path = wheel_house.joinpath(f"{name}-{version}-{wheel_tag}.whl")
+        wheel_builder.save_to(wheel_path)
+        return wheel_path
+
+    return _make_fake_wheel
+
+
 def test_new_resolver_can_install(script):
     create_basic_wheel_for_package(
         script,
@@ -641,8 +661,8 @@ def test_new_resolver_constraint_no_specifier(script):
             "Unnamed requirements are not allowed as constraints",
         ),
         (
-            "req @ https://example.com/dist.zip",
-            "Links are not allowed as constraints",
+            "-e git+https://example.com/dist.git#egg=req",
+            "Editable requirements are not allowed as constraints",
         ),
         (
             "pkg[extra]",
@@ -1278,3 +1298,493 @@ def test_new_resolver_no_fetch_no_satisfying(script):
         "myuberpkg",
     )
     assert "Processing " not in result.stdout, str(result)
+
+
+def test_new_resolver_does_not_install_unneeded_packages_with_url_constraint(script):
+    archive_path = create_basic_wheel_for_package(
+        script,
+        "installed",
+        "0.1.0",
+    )
+    not_installed_path = create_basic_wheel_for_package(
+        script,
+        "not_installed",
+        "0.1.0",
+    )
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("not_installed @ " + path_to_url(not_installed_path))
+
+    (script.scratch_path / "index").mkdir()
+    archive_path.rename(script.scratch_path / "index" / archive_path.name)
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path / "index",
+        "-c", constraints_file,
+        "installed"
+    )
+
+    assert_installed(script, installed="0.1.0")
+    assert_not_installed(script, "not_installed")
+
+
+def test_new_resolver_installs_packages_with_url_constraint(script):
+    installed_path = create_basic_wheel_for_package(
+        script,
+        "installed",
+        "0.1.0",
+    )
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("installed @ " + path_to_url(installed_path))
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-c", constraints_file,
+        "installed"
+    )
+
+    assert_installed(script, installed="0.1.0")
+
+
+def test_new_resolver_reinstall_link_requirement_with_constraint(script):
+    installed_path = create_basic_wheel_for_package(
+        script,
+        "installed",
+        "0.1.0",
+    )
+
+    cr_file = script.scratch_path / "constraints.txt"
+    cr_file.write_text("installed @ " + path_to_url(installed_path))
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-r", cr_file,
+    )
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-c", cr_file,
+        "-r", cr_file,
+    )
+    # TODO: strengthen assertion to "second invocation does no work"
+    # I don't think this is true yet, but it should be in the future.
+
+    assert_installed(script, installed="0.1.0")
+
+
+def test_new_resolver_prefers_url_constraint(script):
+    installed_path = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.1.0",
+    )
+    not_installed_path = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.2.0",
+    )
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("test_pkg @ " + path_to_url(installed_path))
+
+    (script.scratch_path / "index").mkdir()
+    not_installed_path.rename(script.scratch_path / "index" / not_installed_path.name)
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path / "index",
+        "-c", constraints_file,
+        "test_pkg"
+    )
+
+    assert_installed(script, test_pkg="0.1.0")
+
+
+def test_new_resolver_prefers_url_constraint_on_update(script):
+    installed_path = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.1.0",
+    )
+    not_installed_path = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.2.0",
+    )
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("test_pkg @ " + path_to_url(installed_path))
+
+    (script.scratch_path / "index").mkdir()
+    not_installed_path.rename(script.scratch_path / "index" / not_installed_path.name)
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path / "index",
+        "test_pkg"
+    )
+
+    assert_installed(script, test_pkg="0.2.0")
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path / "index",
+        "-c", constraints_file,
+        "test_pkg"
+    )
+
+    assert_installed(script, test_pkg="0.1.0")
+
+
+@pytest.mark.parametrize("version_option", ["--constraint", "--requirement"])
+def test_new_resolver_fails_with_url_constraint_and_incompatible_version(
+    script, version_option,
+):
+    not_installed_path = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.1.0",
+    )
+    not_installed_path = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.2.0",
+    )
+
+    url_constraint = script.scratch_path / "constraints.txt"
+    url_constraint.write_text("test_pkg @ " + path_to_url(not_installed_path))
+
+    version_req = script.scratch_path / "requirements.txt"
+    version_req.write_text("test_pkg<0.2.0")
+
+    result = script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        "--constraint", url_constraint,
+        version_option, version_req,
+        "test_pkg",
+        expect_error=True,
+    )
+
+    assert "Cannot install test_pkg" in result.stderr, str(result)
+    assert (
+        "because these package versions have conflicting dependencies."
+    ) in result.stderr, str(result)
+
+    assert_not_installed(script, "test_pkg")
+
+    # Assert that pip works properly in the absence of the constraints file.
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        version_option, version_req,
+        "test_pkg"
+    )
+
+
+def test_new_resolver_ignores_unneeded_conflicting_constraints(script):
+    version_1 = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.1.0",
+    )
+    version_2 = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.2.0",
+    )
+    create_basic_wheel_for_package(
+        script,
+        "installed",
+        "0.1.0",
+    )
+
+    constraints = [
+        "test_pkg @ " + path_to_url(version_1),
+        "test_pkg @ " + path_to_url(version_2),
+    ]
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("\n".join(constraints))
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        "-c", constraints_file,
+        "installed"
+    )
+
+    assert_not_installed(script, "test_pkg")
+    assert_installed(script, installed="0.1.0")
+
+
+def test_new_resolver_fails_on_needed_conflicting_constraints(script):
+    version_1 = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.1.0",
+    )
+    version_2 = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.2.0",
+    )
+
+    constraints = [
+        "test_pkg @ " + path_to_url(version_1),
+        "test_pkg @ " + path_to_url(version_2),
+    ]
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("\n".join(constraints))
+
+    result = script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        "-c", constraints_file,
+        "test_pkg",
+        expect_error=True,
+    )
+
+    assert (
+        "Cannot install test_pkg because these package versions have conflicting "
+        "dependencies."
+    ) in result.stderr, str(result)
+
+    assert_not_installed(script, "test_pkg")
+
+    # Assert that pip works properly in the absence of the constraints file.
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        "test_pkg",
+    )
+
+
+def test_new_resolver_fails_on_conflicting_constraint_and_requirement(script):
+    version_1 = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.1.0",
+    )
+    version_2 = create_basic_wheel_for_package(
+        script,
+        "test_pkg",
+        "0.2.0",
+    )
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("test_pkg @ " + path_to_url(version_1))
+
+    result = script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        "-c", constraints_file,
+        "test_pkg @ " + path_to_url(version_2),
+        expect_error=True,
+    )
+
+    assert "Cannot install test-pkg 0.2.0" in result.stderr, str(result)
+    assert (
+        "because these package versions have conflicting dependencies."
+    ) in result.stderr, str(result)
+
+    assert_not_installed(script, "test_pkg")
+
+    # Assert that pip works properly in the absence of the constraints file.
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "--find-links", script.scratch_path,
+        "test_pkg @ " + path_to_url(version_2),
+    )
+
+
+@pytest.mark.parametrize("editable", [False, True])
+def test_new_resolver_succeeds_on_matching_constraint_and_requirement(script, editable):
+    if editable:
+        source_dir = create_test_package_with_setup(
+            script,
+            name="test_pkg",
+            version="0.1.0"
+        )
+    else:
+        source_dir = create_basic_wheel_for_package(
+            script,
+            "test_pkg",
+            "0.1.0",
+        )
+
+    req_line = "test_pkg @ " + path_to_url(source_dir)
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text(req_line)
+
+    if editable:
+        last_args = ("-e", source_dir)
+    else:
+        last_args = (req_line,)
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-c", constraints_file,
+        *last_args,
+    )
+
+    assert_installed(script, test_pkg="0.1.0")
+    if editable:
+        assert_editable(script, "test-pkg")
+
+
+def test_new_resolver_applies_url_constraint_to_dep(script):
+    version_1 = create_basic_wheel_for_package(
+        script,
+        "dep",
+        "0.1.0",
+    )
+    version_2 = create_basic_wheel_for_package(
+        script,
+        "dep",
+        "0.2.0",
+    )
+
+    base = create_basic_wheel_for_package(script, "base", "0.1.0", depends=["dep"])
+
+    (script.scratch_path / "index").mkdir()
+    base.rename(script.scratch_path / "index" / base.name)
+    version_2.rename(script.scratch_path / "index" / version_2.name)
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("dep @ " + path_to_url(version_1))
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-c", constraints_file,
+        "--find-links", script.scratch_path / "index",
+        "base",
+    )
+
+    assert_installed(script, dep="0.1.0")
+
+
+def test_new_resolver_handles_compatible_wheel_tags_in_constraint_url(
+    script, make_fake_wheel
+):
+    initial_path = make_fake_wheel("base", "0.1.0", "fakepy1-fakeabi-fakeplat")
+
+    constrained = script.scratch_path / "constrained"
+    constrained.mkdir()
+
+    final_path = constrained / initial_path.name
+
+    initial_path.rename(final_path)
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("base @ " + path_to_url(final_path))
+
+    result = script.pip(
+        "install",
+        "--implementation", "fakepy",
+        '--only-binary=:all:',
+        "--python-version", "1",
+        "--abi", "fakeabi",
+        "--platform", "fakeplat",
+        "--target", script.scratch_path / "target",
+        "--no-cache-dir", "--no-index",
+        "-c", constraints_file,
+        "base",
+    )
+
+    dist_info = Path("scratch", "target", "base-0.1.0.dist-info")
+    result.did_create(dist_info)
+
+
+def test_new_resolver_handles_incompatible_wheel_tags_in_constraint_url(
+    script, make_fake_wheel
+):
+    initial_path = make_fake_wheel("base", "0.1.0", "fakepy1-fakeabi-fakeplat")
+
+    constrained = script.scratch_path / "constrained"
+    constrained.mkdir()
+
+    final_path = constrained / initial_path.name
+
+    initial_path.rename(final_path)
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("base @ " + path_to_url(final_path))
+
+    result = script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-c", constraints_file,
+        "base",
+        expect_error=True,
+    )
+
+    assert (
+        "Cannot install base because these package versions have conflicting "
+        "dependencies."
+    ) in result.stderr, str(result)
+
+    assert_not_installed(script, "base")
+
+
+def test_new_resolver_avoids_incompatible_wheel_tags_in_constraint_url(
+    script, make_fake_wheel
+):
+    initial_path = make_fake_wheel("dep", "0.1.0", "fakepy1-fakeabi-fakeplat")
+
+    constrained = script.scratch_path / "constrained"
+    constrained.mkdir()
+
+    final_path = constrained / initial_path.name
+
+    initial_path.rename(final_path)
+
+    constraints_file = script.scratch_path / "constraints.txt"
+    constraints_file.write_text("dep @ " + path_to_url(final_path))
+
+    index = script.scratch_path / "index"
+    index.mkdir()
+
+    index_dep = create_basic_wheel_for_package(script, "dep", "0.2.0")
+
+    base = create_basic_wheel_for_package(
+        script, "base", "0.1.0"
+    )
+    base_2 = create_basic_wheel_for_package(
+        script, "base", "0.2.0", depends=["dep"]
+    )
+
+    index_dep.rename(index / index_dep.name)
+    base.rename(index / base.name)
+    base_2.rename(index / base_2.name)
+
+    script.pip(
+        "install",
+        "--no-cache-dir", "--no-index",
+        "-c", constraints_file,
+        "--find-links", script.scratch_path / "index",
+        "base",
+    )
+
+    assert_installed(script, base="0.1.0")
+    assert_not_installed(script, "dep")
diff --git a/tests/functional/test_new_resolver_hashes.py b/tests/functional/test_new_resolver_hashes.py
index 854b66418ae..02397616ab7 100644
--- a/tests/functional/test_new_resolver_hashes.py
+++ b/tests/functional/test_new_resolver_hashes.py
@@ -1,7 +1,9 @@
 import collections
 import hashlib
+import json
 
 import pytest
+from pip._vendor.packaging.utils import canonicalize_name
 
 from pip._internal.utils.urls import path_to_url
 from tests.lib import create_basic_sdist_for_package, create_basic_wheel_for_package
@@ -11,6 +13,30 @@
 )
 
 
+def assert_installed(script, **kwargs):
+    ret = script.pip('list', '--format=json')
+    installed = set(
+        (canonicalize_name(val['name']), val['version'])
+        for val in json.loads(ret.stdout)
+    )
+    expected = set((canonicalize_name(k), v) for k, v in kwargs.items())
+    assert expected <= installed, \
+        "{!r} not all in {!r}".format(expected, installed)
+
+
+def assert_not_installed(script, *args):
+    ret = script.pip("list", "--format=json")
+    installed = set(
+        canonicalize_name(val["name"])
+        for val in json.loads(ret.stdout)
+    )
+    # None of the given names should be listed as installed, i.e. their
+    # intersection should be empty.
+    expected = set(canonicalize_name(k) for k in args)
+    assert not (expected & installed), \
+        "{!r} contained in {!r}".format(expected, installed)
+
+
 def _create_find_links(script):
     sdist_path = create_basic_sdist_for_package(script, "base", "0.1.0")
     wheel_path = create_basic_wheel_for_package(script, "base", "0.1.0")
@@ -204,3 +230,81 @@ def test_new_resolver_hash_intersect_empty_from_constraint(script):
         "from some requirements."
     )
     assert message in result.stderr, str(result)
+
+
+@pytest.mark.parametrize("constrain_by_hash", [False, True])
+def test_new_resolver_hash_requirement_and_url_constraint_can_succeed(
+    script, constrain_by_hash,
+):
+    wheel_path = create_basic_wheel_for_package(script, "base", "0.1.0")
+
+    wheel_hash = hashlib.sha256(wheel_path.read_bytes()).hexdigest()
+
+    requirements_txt = script.scratch_path / "requirements.txt"
+    requirements_txt.write_text(
+        """
+        base==0.1.0 --hash=sha256:{wheel_hash}
+        """.format(
+            wheel_hash=wheel_hash,
+        ),
+    )
+
+    constraints_txt = script.scratch_path / "constraints.txt"
+    constraint_text = "base @ {wheel_url}\n".format(wheel_url=path_to_url(wheel_path))
+    if constrain_by_hash:
+        constraint_text += "base==0.1.0 --hash=sha256:{wheel_hash}\n".format(
+            wheel_hash=wheel_hash,
+        )
+    constraints_txt.write_text(constraint_text)
+
+    script.pip(
+        "install",
+        "--no-cache-dir",
+        "--no-index",
+        "--constraint", constraints_txt,
+        "--requirement", requirements_txt,
+    )
+
+    assert_installed(script, base="0.1.0")
+
+
+@pytest.mark.parametrize("constrain_by_hash", [False, True])
+def test_new_resolver_hash_requirement_and_url_constraint_can_fail(
+    script, constrain_by_hash,
+):
+    wheel_path = create_basic_wheel_for_package(script, "base", "0.1.0")
+    other_path = create_basic_wheel_for_package(script, "other", "0.1.0")
+
+    other_hash = hashlib.sha256(other_path.read_bytes()).hexdigest()
+
+    requirements_txt = script.scratch_path / "requirements.txt"
+    requirements_txt.write_text(
+        """
+        base==0.1.0 --hash=sha256:{other_hash}
+        """.format(
+            other_hash=other_hash,
+        ),
+    )
+
+    constraints_txt = script.scratch_path / "constraints.txt"
+    constraint_text = "base @ {wheel_url}\n".format(wheel_url=path_to_url(wheel_path))
+    if constrain_by_hash:
+        constraint_text += "base==0.1.0 --hash=sha256:{other_hash}\n".format(
+            other_hash=other_hash,
+        )
+    constraints_txt.write_text(constraint_text)
+
+    result = script.pip(
+        "install",
+        "--no-cache-dir",
+        "--no-index",
+        "--constraint", constraints_txt,
+        "--requirement", requirements_txt,
+        expect_error=True,
+    )
+
+    assert (
+        "THESE PACKAGES DO NOT MATCH THE HASHES FROM THE REQUIREMENTS FILE."
+    ) in result.stderr, str(result)
+
+    assert_not_installed(script, "base", "other")