diff --git a/src/sphinx_autodoc_typehints/__init__.py b/src/sphinx_autodoc_typehints/__init__.py
index dedfdbe..e621592 100644
--- a/src/sphinx_autodoc_typehints/__init__.py
+++ b/src/sphinx_autodoc_typehints/__init__.py
@@ -36,7 +36,23 @@
     from sphinx.ext.autodoc import Options
 
 _LOGGER = logging.getLogger(__name__)
-_PYDATA_ANNOTATIONS = {"Any", "AnyStr", "Callable", "ClassVar", "Literal", "NoReturn", "Optional", "Tuple", "Union"}
+_PYDATA_ANNOTS_TYPING = {"Any", "AnyStr", "Callable", "ClassVar", "Literal", "NoReturn", "Optional", "Tuple", "Union"}
+_PYDATA_ANNOTS_TYPES = {
+    *("AsyncGeneratorType", "BuiltinFunctionType", "BuiltinMethodType"),
+    *("CellType", "ClassMethodDescriptorType", "CoroutineType"),
+    "EllipsisType",
+    *("FrameType", "FunctionType"),
+    *("GeneratorType", "GetSetDescriptorType"),
+    "LambdaType",
+    *("MemberDescriptorType", "MethodDescriptorType", "MethodType", "MethodWrapperType"),
+    # NoneType is special, but included here for completeness' sake
+    *("NoneType", "NotImplementedType"),
+    "WrapperDescriptorType",
+}
+_PYDATA_ANNOTATIONS = {
+    *(("typing", n) for n in _PYDATA_ANNOTS_TYPING),
+    *(("types", n) for n in _PYDATA_ANNOTS_TYPES),
+}
 
 # types has a bunch of things like ModuleType where ModuleType.__module__ is
 # "builtins" and ModuleType.__name__ is "module", so we have to check for this.
@@ -219,7 +235,7 @@ def format_annotation(annotation: Any, config: Config) -> str:  # noqa: C901, PL
     full_name = f"{module}.{class_name}" if module != "builtins" else class_name
     fully_qualified: bool = getattr(config, "typehints_fully_qualified", False)
     prefix = "" if fully_qualified or full_name == class_name else "~"
-    role = "data" if module == "typing" and class_name in _PYDATA_ANNOTATIONS else "class"
+    role = "data" if (module, class_name) in _PYDATA_ANNOTATIONS else "class"
     args_format = "\\[{}]"
     formatted_args: str | None = ""
 
diff --git a/tests/test_sphinx_autodoc_typehints.py b/tests/test_sphinx_autodoc_typehints.py
index 92c96f4..65c3f6c 100644
--- a/tests/test_sphinx_autodoc_typehints.py
+++ b/tests/test_sphinx_autodoc_typehints.py
@@ -9,7 +9,7 @@
 from io import StringIO
 from pathlib import Path
 from textwrap import dedent, indent
-from types import FunctionType, ModuleType
+from types import EllipsisType, FrameType, FunctionType, ModuleType, NotImplementedType, TracebackType
 from typing import (  # noqa: UP035
     IO,
     Any,
@@ -168,8 +168,12 @@ def test_parse_annotation(annotation: Any, module: str, class_name: str, args: t
     pytest.param(str, ":py:class:`str`", id="str"),
     pytest.param(int, ":py:class:`int`", id="int"),
     pytest.param(StringIO, ":py:class:`~io.StringIO`", id="StringIO"),
-    pytest.param(FunctionType, ":py:class:`~types.FunctionType`", id="FunctionType"),
+    pytest.param(EllipsisType, ":py:data:`~types.EllipsisType`", id="EllipsisType"),
+    pytest.param(FunctionType, ":py:data:`~types.FunctionType`", id="FunctionType"),
+    pytest.param(FrameType, ":py:data:`~types.FrameType`", id="FrameType"),
     pytest.param(ModuleType, ":py:class:`~types.ModuleType`", id="ModuleType"),
+    pytest.param(NotImplementedType, ":py:data:`~types.NotImplementedType`", id="NotImplementedType"),
+    pytest.param(TracebackType, ":py:class:`~types.TracebackType`", id="TracebackType"),
     pytest.param(type(None), ":py:obj:`None`", id="type None"),
     pytest.param(type, ":py:class:`type`", id="type"),
     pytest.param(Callable, ":py:class:`~collections.abc.Callable`", id="abc-Callable"),
@@ -414,7 +418,7 @@ def test_format_annotation(inv: Inventory, annotation: Any, expected_result: str
         assert format_annotation(annotation, conf) == expected_result
 
     # Test for the correct role (class vs data) using the official Sphinx inventory
-    if "typing" in expected_result:
+    if any(modname in expected_result for modname in ("typing", "types")):
         m = re.match(r"^:py:(?P<role>class|data|func):`~(?P<name>[^`]+)`", result)
         assert m, "No match"
         name = m.group("name")