diff --git a/CHANGES b/CHANGES
index 9617665c1..c1946d51d 100644
--- a/CHANGES
+++ b/CHANGES
@@ -21,6 +21,9 @@ Deprecated
 * ``sphinx.roles.Index``
 * ``sphinx.util.detect_encoding()``
 * ``sphinx.util.get_module_source()``
+* ``sphinx.util.inspect.Signature.format_annotation()``
+* ``sphinx.util.inspect.Signature.format_annotation_new()``
+* ``sphinx.util.inspect.Signature.format_annotation_old()``
 
 Features added
 --------------
diff --git a/doc/extdev/deprecated.rst b/doc/extdev/deprecated.rst
index 36b33e829..ec6db2c16 100644
--- a/doc/extdev/deprecated.rst
+++ b/doc/extdev/deprecated.rst
@@ -81,6 +81,21 @@ The following is a list of deprecated interfaces.
      - 4.0
      - N/A
 
+   * - ``sphinx.util.inspect.Signature.format_annotation()``
+     - 2.4
+     - 4.0
+     - ``sphinx.util.typing.stringify()``
+
+   * - ``sphinx.util.inspect.Signature.format_annotation_new()``
+     - 2.4
+     - 4.0
+     - ``sphinx.util.typing.stringify()``
+
+   * - ``sphinx.util.inspect.Signature.format_annotation_old()``
+     - 2.4
+     - 4.0
+     - ``sphinx.util.typing.stringify()``
+
    * - ``sphinx.builders.gettext.POHEADER``
      - 2.3
      - 4.0
diff --git a/sphinx/util/inspect.py b/sphinx/util/inspect.py
index 0b55a92bd..967a10d51 100644
--- a/sphinx/util/inspect.py
+++ b/sphinx/util/inspect.py
@@ -22,9 +22,9 @@ from inspect import (  # NOQA
 from io import StringIO
 from typing import Any, Callable, Mapping, List, Tuple
 
-from sphinx.deprecation import RemovedInSphinx30Warning
+from sphinx.deprecation import RemovedInSphinx30Warning, RemovedInSphinx40Warning
 from sphinx.util import logging
-from sphinx.util.typing import NoneType
+from sphinx.util.typing import stringify as stringify_annotation
 
 if sys.version_info > (3, 7):
     from types import (
@@ -403,11 +403,11 @@ class Signature:
             return None
 
     def format_args(self, show_annotation: bool = True) -> str:
-        def format_param_annotation(param: inspect.Parameter) -> str:
+        def get_annotation(param: inspect.Parameter) -> Any:
             if isinstance(param.annotation, str) and param.name in self.annotations:
-                return self.format_annotation(self.annotations[param.name])
+                return self.annotations[param.name]
             else:
-                return self.format_annotation(param.annotation)
+                return param.annotation
 
         args = []
         last_kind = None
@@ -431,7 +431,7 @@ class Signature:
                 arg.write(param.name)
                 if show_annotation and param.annotation is not param.empty:
                     arg.write(': ')
-                    arg.write(format_param_annotation(param))
+                    arg.write(stringify_annotation(get_annotation(param)))
                 if param.default is not param.empty:
                     if param.annotation is param.empty or show_annotation is False:
                         arg.write('=')
@@ -444,13 +444,13 @@ class Signature:
                 arg.write(param.name)
                 if show_annotation and param.annotation is not param.empty:
                     arg.write(': ')
-                    arg.write(format_param_annotation(param))
+                    arg.write(stringify_annotation(get_annotation(param)))
             elif param.kind == param.VAR_KEYWORD:
                 arg.write('**')
                 arg.write(param.name)
                 if show_annotation and param.annotation is not param.empty:
                     arg.write(': ')
-                    arg.write(format_param_annotation(param))
+                    arg.write(stringify_annotation(get_annotation(param)))
 
             args.append(arg.getvalue())
             last_kind = param.kind
@@ -459,164 +459,29 @@ class Signature:
             return '(%s)' % ', '.join(args)
         else:
             if 'return' in self.annotations:
-                annotation = self.format_annotation(self.annotations['return'])
+                annotation = stringify_annotation(self.annotations['return'])
             else:
-                annotation = self.format_annotation(self.return_annotation)
+                annotation = stringify_annotation(self.return_annotation)
 
             return '(%s) -> %s' % (', '.join(args), annotation)
 
     def format_annotation(self, annotation: Any) -> str:
-        """Return formatted representation of a type annotation.
-
-        Show qualified names for types and additional details for types from
-        the ``typing`` module.
-
-        Displaying complex types from ``typing`` relies on its private API.
-        """
-        if isinstance(annotation, str):
-            return annotation
-        elif isinstance(annotation, typing.TypeVar):  # type: ignore
-            return annotation.__name__
-        elif not annotation:
-            return repr(annotation)
-        elif annotation is NoneType:  # type: ignore
-            return 'None'
-        elif getattr(annotation, '__module__', None) == 'builtins':
-            return annotation.__qualname__
-        elif annotation is Ellipsis:
-            return '...'
-
-        if sys.version_info >= (3, 7):  # py37+
-            return self.format_annotation_new(annotation)
-        else:
-            return self.format_annotation_old(annotation)
+        """Return formatted representation of a type annotation."""
+        warnings.warn('format_annotation() is deprecated',
+                      RemovedInSphinx40Warning)
+        return stringify_annotation(annotation)
 
     def format_annotation_new(self, annotation: Any) -> str:
         """format_annotation() for py37+"""
-        module = getattr(annotation, '__module__', None)
-        if module == 'typing':
-            if getattr(annotation, '_name', None):
-                qualname = annotation._name
-            elif getattr(annotation, '__qualname__', None):
-                qualname = annotation.__qualname__
-            elif getattr(annotation, '__forward_arg__', None):
-                qualname = annotation.__forward_arg__
-            else:
-                qualname = self.format_annotation(annotation.__origin__)  # ex. Union
-        elif hasattr(annotation, '__qualname__'):
-            qualname = '%s.%s' % (module, annotation.__qualname__)
-        else:
-            qualname = repr(annotation)
-
-        if getattr(annotation, '__args__', None):
-            if qualname == 'Union':
-                if len(annotation.__args__) == 2 and annotation.__args__[1] is NoneType:  # type: ignore  # NOQA
-                    return 'Optional[%s]' % self.format_annotation(annotation.__args__[0])
-                else:
-                    args = ', '.join(self.format_annotation(a) for a in annotation.__args__)
-                    return '%s[%s]' % (qualname, args)
-            elif qualname == 'Callable':
-                args = ', '.join(self.format_annotation(a) for a in annotation.__args__[:-1])
-                returns = self.format_annotation(annotation.__args__[-1])
-                return '%s[[%s], %s]' % (qualname, args, returns)
-            elif annotation._special:
-                return qualname
-            else:
-                args = ', '.join(self.format_annotation(a) for a in annotation.__args__)
-                return '%s[%s]' % (qualname, args)
-
-        return qualname
+        warnings.warn('format_annotation_new() is deprecated',
+                      RemovedInSphinx40Warning)
+        return stringify_annotation(annotation)
 
     def format_annotation_old(self, annotation: Any) -> str:
         """format_annotation() for py36 or below"""
-        module = getattr(annotation, '__module__', None)
-        if module == 'typing':
-            if getattr(annotation, '_name', None):
-                qualname = annotation._name
-            elif getattr(annotation, '__qualname__', None):
-                qualname = annotation.__qualname__
-            elif getattr(annotation, '__forward_arg__', None):
-                qualname = annotation.__forward_arg__
-            elif getattr(annotation, '__origin__', None):
-                qualname = self.format_annotation(annotation.__origin__)  # ex. Union
-            else:
-                qualname = repr(annotation).replace('typing.', '')
-        elif hasattr(annotation, '__qualname__'):
-            qualname = '%s.%s' % (module, annotation.__qualname__)
-        else:
-            qualname = repr(annotation)
-
-        if (isinstance(annotation, typing.TupleMeta) and  # type: ignore
-                not hasattr(annotation, '__tuple_params__')):  # for Python 3.6
-            params = annotation.__args__
-            if params:
-                param_str = ', '.join(self.format_annotation(p) for p in params)
-                return '%s[%s]' % (qualname, param_str)
-            else:
-                return qualname
-        elif isinstance(annotation, typing.GenericMeta):
-            params = None
-            if hasattr(annotation, '__args__'):
-                # for Python 3.5.2+
-                if annotation.__args__ is None or len(annotation.__args__) <= 2:  # type: ignore  # NOQA
-                    params = annotation.__args__  # type: ignore
-                else:  # typing.Callable
-                    args = ', '.join(self.format_annotation(arg) for arg
-                                     in annotation.__args__[:-1])  # type: ignore
-                    result = self.format_annotation(annotation.__args__[-1])  # type: ignore
-                    return '%s[[%s], %s]' % (qualname, args, result)
-            elif hasattr(annotation, '__parameters__'):
-                # for Python 3.5.0 and 3.5.1
-                params = annotation.__parameters__  # type: ignore
-            if params is not None:
-                param_str = ', '.join(self.format_annotation(p) for p in params)
-                return '%s[%s]' % (qualname, param_str)
-        elif (hasattr(typing, 'UnionMeta') and
-              isinstance(annotation, typing.UnionMeta) and  # type: ignore
-              hasattr(annotation, '__union_params__')):  # for Python 3.5
-            params = annotation.__union_params__
-            if params is not None:
-                if len(params) == 2 and params[1] is NoneType:  # type: ignore
-                    return 'Optional[%s]' % self.format_annotation(params[0])
-                else:
-                    param_str = ', '.join(self.format_annotation(p) for p in params)
-                    return '%s[%s]' % (qualname, param_str)
-        elif (hasattr(annotation, '__origin__') and
-              annotation.__origin__ is typing.Union):  # for Python 3.5.2+
-            params = annotation.__args__
-            if params is not None:
-                if len(params) == 2 and params[1] is NoneType:  # type: ignore
-                    return 'Optional[%s]' % self.format_annotation(params[0])
-                else:
-                    param_str = ', '.join(self.format_annotation(p) for p in params)
-                    return 'Union[%s]' % param_str
-        elif (isinstance(annotation, typing.CallableMeta) and  # type: ignore
-              getattr(annotation, '__args__', None) is not None and
-              hasattr(annotation, '__result__')):  # for Python 3.5
-            # Skipped in the case of plain typing.Callable
-            args = annotation.__args__
-            if args is None:
-                return qualname
-            elif args is Ellipsis:
-                args_str = '...'
-            else:
-                formatted_args = (self.format_annotation(a) for a in args)
-                args_str = '[%s]' % ', '.join(formatted_args)
-            return '%s[%s, %s]' % (qualname,
-                                   args_str,
-                                   self.format_annotation(annotation.__result__))
-        elif (isinstance(annotation, typing.TupleMeta) and  # type: ignore
-              hasattr(annotation, '__tuple_params__') and
-              hasattr(annotation, '__tuple_use_ellipsis__')):  # for Python 3.5
-            params = annotation.__tuple_params__
-            if params is not None:
-                param_strings = [self.format_annotation(p) for p in params]
-                if annotation.__tuple_use_ellipsis__:
-                    param_strings.append('...')
-                return '%s[%s]' % (qualname,
-                                   ', '.join(param_strings))
-
-        return qualname
+        warnings.warn('format_annotation_old() is deprecated',
+                      RemovedInSphinx40Warning)
+        return stringify_annotation(annotation)
 
 
 def getdoc(obj: Any, attrgetter: Callable = safe_getattr,
diff --git a/sphinx/util/typing.py b/sphinx/util/typing.py
index 1b2ec3f60..ccceefed6 100644
--- a/sphinx/util/typing.py
+++ b/sphinx/util/typing.py
@@ -8,7 +8,9 @@
     :license: BSD, see LICENSE for details.
 """
 
-from typing import Any, Callable, Dict, List, Tuple, Union
+import sys
+import typing
+from typing import Any, Callable, Dict, List, Tuple, TypeVar, Union
 
 from docutils import nodes
 from docutils.parsers.rst.states import Inliner
@@ -35,3 +37,153 @@ TitleGetter = Callable[[nodes.Node], str]
 
 # inventory data on memory
 Inventory = Dict[str, Dict[str, Tuple[str, str, str, str]]]
+
+
+def stringify(annotation: Any) -> str:
+    """Stringify type annotation object."""
+    if isinstance(annotation, str):
+        return annotation
+    elif isinstance(annotation, TypeVar):  # type: ignore
+        return annotation.__name__
+    elif not annotation:
+        return repr(annotation)
+    elif annotation is NoneType:  # type: ignore
+        return 'None'
+    elif getattr(annotation, '__module__', None) == 'builtins':
+        return annotation.__qualname__
+    elif annotation is Ellipsis:
+        return '...'
+
+    if sys.version_info >= (3, 7):  # py37+
+        return _stringify_py37(annotation)
+    else:
+        return _stringify_py36(annotation)
+
+
+def _stringify_py37(annotation: Any) -> str:
+    """stringify() for py37+."""
+    module = getattr(annotation, '__module__', None)
+    if module == 'typing':
+        if getattr(annotation, '_name', None):
+            qualname = annotation._name
+        elif getattr(annotation, '__qualname__', None):
+            qualname = annotation.__qualname__
+        elif getattr(annotation, '__forward_arg__', None):
+            qualname = annotation.__forward_arg__
+        else:
+            qualname = stringify(annotation.__origin__)  # ex. Union
+    elif hasattr(annotation, '__qualname__'):
+        qualname = '%s.%s' % (module, annotation.__qualname__)
+    else:
+        qualname = repr(annotation)
+
+    if getattr(annotation, '__args__', None):
+        if qualname == 'Union':
+            if len(annotation.__args__) == 2 and annotation.__args__[1] is NoneType:  # type: ignore  # NOQA
+                return 'Optional[%s]' % stringify(annotation.__args__[0])
+            else:
+                args = ', '.join(stringify(a) for a in annotation.__args__)
+                return '%s[%s]' % (qualname, args)
+        elif qualname == 'Callable':
+            args = ', '.join(stringify(a) for a in annotation.__args__[:-1])
+            returns = stringify(annotation.__args__[-1])
+            return '%s[[%s], %s]' % (qualname, args, returns)
+        elif annotation._special:
+            return qualname
+        else:
+            args = ', '.join(stringify(a) for a in annotation.__args__)
+            return '%s[%s]' % (qualname, args)
+
+    return qualname
+
+
+def _stringify_py36(annotation: Any) -> str:
+    """stringify() for py35 and py36."""
+    module = getattr(annotation, '__module__', None)
+    if module == 'typing':
+        if getattr(annotation, '_name', None):
+            qualname = annotation._name
+        elif getattr(annotation, '__qualname__', None):
+            qualname = annotation.__qualname__
+        elif getattr(annotation, '__forward_arg__', None):
+            qualname = annotation.__forward_arg__
+        elif getattr(annotation, '__origin__', None):
+            qualname = stringify(annotation.__origin__)  # ex. Union
+        else:
+            qualname = repr(annotation).replace('typing.', '')
+    elif hasattr(annotation, '__qualname__'):
+        qualname = '%s.%s' % (module, annotation.__qualname__)
+    else:
+        qualname = repr(annotation)
+
+    if (isinstance(annotation, typing.TupleMeta) and  # type: ignore
+            not hasattr(annotation, '__tuple_params__')):  # for Python 3.6
+        params = annotation.__args__
+        if params:
+            param_str = ', '.join(stringify(p) for p in params)
+            return '%s[%s]' % (qualname, param_str)
+        else:
+            return qualname
+    elif isinstance(annotation, typing.GenericMeta):
+        params = None
+        if hasattr(annotation, '__args__'):
+            # for Python 3.5.2+
+            if annotation.__args__ is None or len(annotation.__args__) <= 2:  # type: ignore  # NOQA
+                params = annotation.__args__  # type: ignore
+            else:  # typing.Callable
+                args = ', '.join(stringify(arg) for arg
+                                 in annotation.__args__[:-1])  # type: ignore
+                result = stringify(annotation.__args__[-1])  # type: ignore
+                return '%s[[%s], %s]' % (qualname, args, result)
+        elif hasattr(annotation, '__parameters__'):
+            # for Python 3.5.0 and 3.5.1
+            params = annotation.__parameters__  # type: ignore
+        if params is not None:
+            param_str = ', '.join(stringify(p) for p in params)
+            return '%s[%s]' % (qualname, param_str)
+    elif (hasattr(typing, 'UnionMeta') and
+          isinstance(annotation, typing.UnionMeta) and  # type: ignore
+          hasattr(annotation, '__union_params__')):  # for Python 3.5
+        params = annotation.__union_params__
+        if params is not None:
+            if len(params) == 2 and params[1] is NoneType:  # type: ignore
+                return 'Optional[%s]' % stringify(params[0])
+            else:
+                param_str = ', '.join(stringify(p) for p in params)
+                return '%s[%s]' % (qualname, param_str)
+    elif (hasattr(annotation, '__origin__') and
+          annotation.__origin__ is typing.Union):  # for Python 3.5.2+
+        params = annotation.__args__
+        if params is not None:
+            if len(params) == 2 and params[1] is NoneType:  # type: ignore
+                return 'Optional[%s]' % stringify(params[0])
+            else:
+                param_str = ', '.join(stringify(p) for p in params)
+                return 'Union[%s]' % param_str
+    elif (isinstance(annotation, typing.CallableMeta) and  # type: ignore
+          getattr(annotation, '__args__', None) is not None and
+          hasattr(annotation, '__result__')):  # for Python 3.5
+        # Skipped in the case of plain typing.Callable
+        args = annotation.__args__
+        if args is None:
+            return qualname
+        elif args is Ellipsis:
+            args_str = '...'
+        else:
+            formatted_args = (stringify(a) for a in args)
+            args_str = '[%s]' % ', '.join(formatted_args)
+        return '%s[%s, %s]' % (qualname,
+                               args_str,
+                               stringify(annotation.__result__))
+    elif (isinstance(annotation, typing.TupleMeta) and  # type: ignore
+          hasattr(annotation, '__tuple_params__') and
+          hasattr(annotation, '__tuple_use_ellipsis__')):  # for Python 3.5
+        params = annotation.__tuple_params__
+        if params is not None:
+            param_strings = [stringify(p) for p in params]
+            if annotation.__tuple_use_ellipsis__:
+                param_strings.append('...')
+            return '%s[%s]' % (qualname,
+                               ', '.join(param_strings))
+
+    return qualname
diff --git a/tests/test_util_typing.py b/tests/test_util_typing.py
new file mode 100644
index 000000000..9a225f0f1
--- /dev/null
+++ b/tests/test_util_typing.py
@@ -0,0 +1,98 @@
+"""
+    test_util_typing
+    ~~~~~~~~~~~~~~~~
+
+    Tests util.typing functions.
+
+    :copyright: Copyright 2007-2019 by the Sphinx team, see AUTHORS.
+    :license: BSD, see LICENSE for details.
+"""
+
+import sys
+from numbers import Integral
+from typing import Any, Dict, List, TypeVar, Union, Callable, Tuple, Optional
+
+from sphinx.util.typing import stringify
+
+
+class MyClass1:
+    pass
+
+
+class MyClass2(MyClass1):
+    __qualname__ = '<MyClass2>'
+
+
+def test_stringify():
+    assert stringify(int) == "int"
+    assert stringify(str) == "str"
+    assert stringify(None) == "None"
+    assert stringify(Integral) == "numbers.Integral"
+    assert stringify(Any) == "Any"
+
+
+def test_stringify_type_hints_containers():
+    assert stringify(List) == "List"
+    assert stringify(Dict) == "Dict"
+    assert stringify(List[int]) == "List[int]"
+    assert stringify(List[str]) == "List[str]"
+    assert stringify(Dict[str, float]) == "Dict[str, float]"
+    assert stringify(Tuple[str, str, str]) == "Tuple[str, str, str]"
+    assert stringify(Tuple[str, ...]) == "Tuple[str, ...]"
+    assert stringify(List[Dict[str, Tuple]]) == "List[Dict[str, Tuple]]"
+
+
+def test_stringify_type_hints_string():
+    assert stringify("int") == "int"
+    assert stringify("str") == "str"
+    assert stringify(List["int"]) == "List[int]"
+    assert stringify("Tuple[str]") == "Tuple[str]"
+    assert stringify("unknown") == "unknown"
+
+
+def test_stringify_type_hints_Callable():
+    assert stringify(Callable) == "Callable"
+
+    if sys.version_info >= (3, 7):
+        assert stringify(Callable[[str], int]) == "Callable[[str], int]"
+        assert stringify(Callable[..., int]) == "Callable[[...], int]"
+    else:
+        assert stringify(Callable[[str], int]) == "Callable[str, int]"
+        assert stringify(Callable[..., int]) == "Callable[..., int]"
+
+
+def test_stringify_type_hints_Union():
+    assert stringify(Optional[int]) == "Optional[int]"
+    assert stringify(Union[str, None]) == "Optional[str]"
+    assert stringify(Union[int, str]) == "Union[int, str]"
+
+    if sys.version_info >= (3, 7):
+        assert stringify(Union[int, Integral]) == "Union[int, numbers.Integral]"
+        assert (stringify(Union[MyClass1, MyClass2]) ==
+                "Union[test_util_typing.MyClass1, test_util_typing.<MyClass2>]")
+    else:
+        assert stringify(Union[int, Integral]) == "numbers.Integral"
+        assert stringify(Union[MyClass1, MyClass2]) == "test_util_typing.MyClass1"
+
+
+def test_stringify_type_hints_typevars():
+    T = TypeVar('T')
+    T_co = TypeVar('T_co', covariant=True)
+    T_contra = TypeVar('T_contra', contravariant=True)
+
+    assert stringify(T) == "T"
+    assert stringify(T_co) == "T_co"
+    assert stringify(T_contra) == "T_contra"
+    assert stringify(List[T]) == "List[T]"
+
+
+def test_stringify_type_hints_custom_class():
+    assert stringify(MyClass1) == "test_util_typing.MyClass1"
+    assert stringify(MyClass2) == "test_util_typing.<MyClass2>"
+
+
+def test_stringify_type_hints_alias():
+    MyStr = str
+    MyTuple = Tuple[str, str]
+    assert stringify(MyStr) == "str"
+    assert stringify(MyTuple) == "Tuple[str, str]"  # type: ignore