diff --git a/setup.py b/setup.py
--- a/setup.py
+++ b/setup.py
@@ -39,6 +39,7 @@
     "Programming Language :: Python :: 3.8",
     "Programming Language :: Python :: 3.9",
     "Programming Language :: Python :: 3.10",
+    "Programming Language :: Python :: 3.11",
     "Programming Language :: Python :: Implementation :: CPython",
     "Programming Language :: Python :: Implementation :: PyPy",
     "Topic :: Software Development :: Libraries :: Python Modules",
diff --git a/src/attr/_compat.py b/src/attr/_compat.py
--- a/src/attr/_compat.py
+++ b/src/attr/_compat.py
@@ -111,12 +111,10 @@ def force_x_to_be_a_cell():  # pragma: no cover
         # Convert this code object to a code object that sets the
         # function's first _freevar_ (not cellvar) to the argument.
         if sys.version_info >= (3, 8):
-            # CPython 3.8+ has an incompatible CodeType signature
-            # (added a posonlyargcount argument) but also added
-            # CodeType.replace() to do this without counting parameters.
-            set_first_freevar_code = co.replace(
-                co_cellvars=co.co_freevars, co_freevars=co.co_cellvars
-            )
+
+            def set_closure_cell(cell, value):
+                cell.cell_contents = value
+
         else:
             args = [co.co_argcount]
             args.append(co.co_kwonlyargcount)
@@ -140,15 +138,15 @@ def force_x_to_be_a_cell():  # pragma: no cover
             )
             set_first_freevar_code = types.CodeType(*args)
 
-        def set_closure_cell(cell, value):
-            # Create a function using the set_first_freevar_code,
-            # whose first closure cell is `cell`. Calling it will
-            # change the value of that cell.
-            setter = types.FunctionType(
-                set_first_freevar_code, {}, "setter", (), (cell,)
-            )
-            # And call it to set the cell.
-            setter(value)
+            def set_closure_cell(cell, value):
+                # Create a function using the set_first_freevar_code,
+                # whose first closure cell is `cell`. Calling it will
+                # change the value of that cell.
+                setter = types.FunctionType(
+                    set_first_freevar_code, {}, "setter", (), (cell,)
+                )
+                # And call it to set the cell.
+                setter(value)
 
         # Make sure it works on this interpreter:
         def make_func_with_cell():

diff --git a/tests/test_annotations.py b/tests/test_annotations.py
--- a/tests/test_annotations.py
+++ b/tests/test_annotations.py
@@ -94,6 +94,10 @@ class C:
         assert 1 == len(attr.fields(C))
         assert_init_annotations(C, x=typing.List[int])
 
+    @pytest.mark.skipif(
+        sys.version_info[:2] < (3, 11),
+        reason="Incompatible behavior on older Pythons",
+    )
     @pytest.mark.parametrize("slots", [True, False])
     def test_auto_attribs(self, slots):
         """
@@ -149,7 +153,7 @@ class C:
             x=typing.List[int],
             y=int,
             z=int,
-            foo=typing.Optional[typing.Any],
+            foo=typing.Any,
         )
 
     @pytest.mark.parametrize("slots", [True, False])
@@ -384,8 +388,9 @@ def noop():
 
         assert attr.converters.optional(noop).__annotations__ == {}
 
-    @pytest.mark.xfail(
-        sys.version_info[:2] == (3, 6), reason="Does not work on 3.6."
+    @pytest.mark.skipif(
+        sys.version_info[:2] < (3, 11),
+        reason="Incompatible behavior on older Pythons",
     )
     @pytest.mark.parametrize("slots", [True, False])
     def test_annotations_strings(self, slots):
@@ -417,7 +422,7 @@ class C:
             x=typing.List[int],
             y=int,
             z=int,
-            foo=typing.Optional[typing.Any],
+            foo=typing.Any,
         )
 
     @pytest.mark.parametrize("slots", [True, False])
diff --git a/tests/test_make.py b/tests/test_make.py
--- a/tests/test_make.py
+++ b/tests/test_make.py
@@ -2275,7 +2275,9 @@ class C:
             def __getstate__(self):
                 return ("hi",)
 
-        assert None is getattr(C(), "__setstate__", None)
+        assert getattr(object, "__setstate__", None) is getattr(
+            C, "__setstate__", None
+        )
 
         @attr.s(slots=slots, auto_detect=True)
         class C:
@@ -2291,7 +2293,9 @@ def __setstate__(self, state):
         i.__setstate__(())
 
         assert True is i.called
-        assert None is getattr(C(), "__getstate__", None)
+        assert getattr(object, "__getstate__", None) is getattr(
+            C, "__getstate__", None
+        )
 
     @pytest.mark.skipif(PY310, reason="Pre-3.10 only.")
     def test_match_args_pre_310(self):
diff --git a/tests/test_slots.py b/tests/test_slots.py
--- a/tests/test_slots.py
+++ b/tests/test_slots.py
@@ -660,10 +660,12 @@ def test_no_getstate_setstate_for_dict_classes(self):
         As long as getstate_setstate is None, nothing is done to dict
         classes.
         """
-        i = C1(1, 2)
-
-        assert None is getattr(i, "__getstate__", None)
-        assert None is getattr(i, "__setstate__", None)
+        assert getattr(object, "__getstate__", None) is getattr(
+            C1, "__getstate__", None
+        )
+        assert getattr(object, "__setstate__", None) is getattr(
+            C1, "__setstate__", None
+        )
 
     def test_no_getstate_setstate_if_option_false(self):
         """
@@ -674,10 +676,12 @@ def test_no_getstate_setstate_if_option_false(self):
         class C:
             x = attr.ib()
 
-        i = C(42)
-
-        assert None is getattr(i, "__getstate__", None)
-        assert None is getattr(i, "__setstate__", None)
+        assert getattr(object, "__getstate__", None) is getattr(
+            C, "__getstate__", None
+        )
+        assert getattr(object, "__setstate__", None) is getattr(
+            C, "__setstate__", None
+        )
 
     @pytest.mark.parametrize("cls", [C2(1), C2Slots(1)])
     def test_getstate_set_state_force_true(self, cls):

Fix set_closure_cell on 3.11
The fix for 3.11 compatibility (https://github.com/python-attrs/attrs/issues/907).

Maybe hold off on merging until a discussion on BPO.

I guess we'll have to finish this up before 3.11? :-/
I don't think the other failing tests were related to this, but other 3.11 incompatibilities. Since the beta is out I can revisit this