diff --git a/Doc/library/tempfile.rst b/Doc/library/tempfile.rst
index 00acf4b1792375..f182c0feaa7e7c 100644
--- a/Doc/library/tempfile.rst
+++ b/Doc/library/tempfile.rst
@@ -105,12 +105,12 @@ The module defines the following user-callable items:
       the truncate method now accepts a ``size`` argument.
 
 
-.. function:: TemporaryDirectory(suffix=None, prefix=None, dir=None)
+.. function:: TemporaryDirectory(suffix=None, prefix=None, dir=None, ignore_cleanup_errors=False)
 
    This function securely creates a temporary directory using the same rules as :func:`mkdtemp`.
    The resulting object can be used as a context manager (see
    :ref:`tempfile-examples`).  On completion of the context or destruction
-   of the temporary directory object the newly created temporary directory
+   of the temporary directory object, the newly created temporary directory
    and all its contents are removed from the filesystem.
 
    The directory name can be retrieved from the :attr:`name` attribute of the
@@ -119,10 +119,19 @@ The module defines the following user-callable items:
    the :keyword:`with` statement, if there is one.
 
    The directory can be explicitly cleaned up by calling the
-   :func:`cleanup` method.
+   :func:`cleanup` method. If *ignore_cleanup_errors* is true, any unhandled
+   exceptions during explicit or implicit cleanup (such as a
+   :exc:`PermissionError` removing open files on Windows) will be ignored,
+   and the remaining removable items deleted on a "best-effort" basis.
+   Otherwise, errors will be raised in whatever context cleanup occurs
+   (the :func:`cleanup` call, exiting the context manager, when the object
+   is garbage-collected or during interpreter shutdown).
 
    .. versionadded:: 3.2
 
+   .. versionchanged:: 3.10
+      Added *ignore_cleanup_errors* parameter.
+
 
 .. function:: mkstemp(suffix=None, prefix=None, dir=None, text=False)
 
diff --git a/Lib/tempfile.py b/Lib/tempfile.py
index f92db5313215e9..2ba92ad9f03466 100644
--- a/Lib/tempfile.py
+++ b/Lib/tempfile.py
@@ -774,7 +774,7 @@ def writelines(self, iterable):
         return rv
 
 
-class TemporaryDirectory(object):
+class TemporaryDirectory:
     """Create and return a temporary directory.  This has the same
     behavior as mkdtemp but can be used as a context manager.  For
     example:
@@ -786,15 +786,49 @@ class TemporaryDirectory(object):
     in it are removed.
     """
 
-    def __init__(self, suffix=None, prefix=None, dir=None):
+    def __init__(self, suffix=None, prefix=None, dir=None,
+                 ignore_cleanup_errors=False):
         self.name = mkdtemp(suffix, prefix, dir)
+        self._ignore_cleanup_errors = ignore_cleanup_errors
         self._finalizer = _weakref.finalize(
             self, self._cleanup, self.name,
-            warn_message="Implicitly cleaning up {!r}".format(self))
+            warn_message="Implicitly cleaning up {!r}".format(self),
+            ignore_errors=self._ignore_cleanup_errors)
 
     @classmethod
-    def _cleanup(cls, name, warn_message):
-        _shutil.rmtree(name)
+    def _rmtree(cls, name, ignore_errors=False):
+        def onerror(func, path, exc_info):
+            if issubclass(exc_info[0], PermissionError):
+                def resetperms(path):
+                    try:
+                        _os.chflags(path, 0)
+                    except AttributeError:
+                        pass
+                    _os.chmod(path, 0o700)
+
+                try:
+                    if path != name:
+                        resetperms(_os.path.dirname(path))
+                    resetperms(path)
+
+                    try:
+                        _os.unlink(path)
+                    # PermissionError is raised on FreeBSD for directories
+                    except (IsADirectoryError, PermissionError):
+                        cls._rmtree(path, ignore_errors=ignore_errors)
+                except FileNotFoundError:
+                    pass
+            elif issubclass(exc_info[0], FileNotFoundError):
+                pass
+            else:
+                if not ignore_errors:
+                    raise
+
+        _shutil.rmtree(name, onerror=onerror)
+
+    @classmethod
+    def _cleanup(cls, name, warn_message, ignore_errors=False):
+        cls._rmtree(name, ignore_errors=ignore_errors)
         _warnings.warn(warn_message, ResourceWarning)
 
     def __repr__(self):
@@ -807,5 +841,7 @@ def __exit__(self, exc, value, tb):
         self.cleanup()
 
     def cleanup(self):
-        if self._finalizer.detach():
-            _shutil.rmtree(self.name)
+        if self._finalizer.detach() or _os.path.exists(self.name):
+            self._rmtree(self.name, ignore_errors=self._ignore_cleanup_errors)
+
+    __class_getitem__ = classmethod(_types.GenericAlias)
diff --git a/Lib/test/test_tempfile.py b/Lib/test/test_tempfile.py
index 2d47e70bd03288..8171aa6e0ae52a 100644
--- a/Lib/test/test_tempfile.py
+++ b/Lib/test/test_tempfile.py
@@ -1312,13 +1312,17 @@ def __exit__(self, *exc_info):
             d.clear()
             d.update(c)
 
+
 class TestTemporaryDirectory(BaseTestCase):
     """Test TemporaryDirectory()."""
 
-    def do_create(self, dir=None, pre="", suf="", recurse=1):
+    def do_create(self, dir=None, pre="", suf="", recurse=1, dirs=1, files=1,
+                  ignore_cleanup_errors=False):
         if dir is None:
             dir = tempfile.gettempdir()
-        tmp = tempfile.TemporaryDirectory(dir=dir, prefix=pre, suffix=suf)
+        tmp = tempfile.TemporaryDirectory(
+            dir=dir, prefix=pre, suffix=suf,
+            ignore_cleanup_errors=ignore_cleanup_errors)
         self.nameCheck(tmp.name, dir, pre, suf)
         # Create a subdirectory and some files
         if recurse:
@@ -1351,7 +1355,31 @@ def test_explicit_cleanup(self):
         finally:
             os.rmdir(dir)
 
-    @support.skip_unless_symlink
+    def test_explict_cleanup_ignore_errors(self):
+        """Test that cleanup doesn't return an error when ignoring them."""
+        with tempfile.TemporaryDirectory() as working_dir:
+            temp_dir = self.do_create(
+                dir=working_dir, ignore_cleanup_errors=True)
+            temp_path = pathlib.Path(temp_dir.name)
+            self.assertTrue(temp_path.exists(),
+                            f"TemporaryDirectory {temp_path!s} does not exist")
+            with open(temp_path / "a_file.txt", "w+t") as open_file:
+                open_file.write("Hello world!\n")
+                temp_dir.cleanup()
+            self.assertEqual(len(list(temp_path.glob("*"))),
+                             int(sys.platform.startswith("win")),
+                             "Unexpected number of files in "
+                             f"TemporaryDirectory {temp_path!s}")
+            self.assertEqual(
+                temp_path.exists(),
+                sys.platform.startswith("win"),
+                f"TemporaryDirectory {temp_path!s} existance state unexpected")
+            temp_dir.cleanup()
+            self.assertFalse(
+                temp_path.exists(),
+                f"TemporaryDirectory {temp_path!s} exists after cleanup")
+
+    @os_helper.skip_unless_symlink
     def test_cleanup_with_symlink_to_a_directory(self):
         # cleanup() should not follow symlinks to directories (issue #12464)
         d1 = self.do_create()
@@ -1385,6 +1413,27 @@ def test_del_on_collection(self):
         finally:
             os.rmdir(dir)
 
+    @support.cpython_only
+    def test_del_on_collection_ignore_errors(self):
+        """Test that ignoring errors works when TemporaryDirectory is gced."""
+        with tempfile.TemporaryDirectory() as working_dir:
+            temp_dir = self.do_create(
+                dir=working_dir, ignore_cleanup_errors=True)
+            temp_path = pathlib.Path(temp_dir.name)
+            self.assertTrue(temp_path.exists(),
+                            f"TemporaryDirectory {temp_path!s} does not exist")
+            with open(temp_path / "a_file.txt", "w+t") as open_file:
+                open_file.write("Hello world!\n")
+                del temp_dir
+            self.assertEqual(len(list(temp_path.glob("*"))),
+                             int(sys.platform.startswith("win")),
+                             "Unexpected number of files in "
+                             f"TemporaryDirectory {temp_path!s}")
+            self.assertEqual(
+                temp_path.exists(),
+                sys.platform.startswith("win"),
+                f"TemporaryDirectory {temp_path!s} existance state unexpected")
+
     def test_del_on_shutdown(self):
         # A TemporaryDirectory may be cleaned up during shutdown
         with self.do_create() as dir:
@@ -1417,6 +1466,43 @@ def test_del_on_shutdown(self):
                 self.assertNotIn("Exception ", err)
                 self.assertIn("ResourceWarning: Implicitly cleaning up", err)
 
+    def test_del_on_shutdown_ignore_errors(self):
+        """Test ignoring errors works when a tempdir is gc'ed on shutdown."""
+        with tempfile.TemporaryDirectory() as working_dir:
+            code = """if True:
+                import pathlib
+                import sys
+                import tempfile
+                import warnings
+
+                temp_dir = tempfile.TemporaryDirectory(
+                    dir={working_dir!r}, ignore_cleanup_errors=True)
+                sys.stdout.buffer.write(temp_dir.name.encode())
+
+                temp_dir_2 = pathlib.Path(temp_dir.name) / "test_dir"
+                temp_dir_2.mkdir()
+                with open(temp_dir_2 / "test0.txt", "w") as test_file:
+                    test_file.write("Hello world!")
+                open_file = open(temp_dir_2 / "open_file.txt", "w")
+                open_file.write("Hello world!")
+
+                warnings.filterwarnings("always", category=ResourceWarning)
+                """.format(working_dir=working_dir)
+            __, out, err = script_helper.assert_python_ok("-c", code)
+            temp_path = pathlib.Path(out.decode().strip())
+            self.assertEqual(len(list(temp_path.glob("*"))),
+                             int(sys.platform.startswith("win")),
+                             "Unexpected number of files in "
+                             f"TemporaryDirectory {temp_path!s}")
+            self.assertEqual(
+                temp_path.exists(),
+                sys.platform.startswith("win"),
+                f"TemporaryDirectory {temp_path!s} existance state unexpected")
+            err = err.decode('utf-8', 'backslashreplace')
+            self.assertNotIn("Exception", err)
+            self.assertNotIn("Error", err)
+            self.assertIn("ResourceWarning: Implicitly cleaning up", err)
+
     def test_exit_on_shutdown(self):
         # Issue #22427
         with self.do_create() as dir:
diff --git a/Misc/NEWS.d/next/Library/2021-03-07-23-23-03.bpo-29982.Q9iszT.rst b/Misc/NEWS.d/next/Library/2021-03-07-23-23-03.bpo-29982.Q9iszT.rst
new file mode 100644
index 00000000000000..fd71bc6e4e0df7
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2021-03-07-23-23-03.bpo-29982.Q9iszT.rst
@@ -0,0 +1,3 @@
+Add optional parameter *ignore_cleanup_errors* to
+:func:`tempfile.TemporaryDirectory` and allow multiple :func:`cleanup` attempts.
+Contributed by C.A.M. Gerlach.