diff --git a/Lib/inspect.py b/Lib/inspect.py
index 1c36579..e2759f4 100644
--- a/Lib/inspect.py
+++ b/Lib/inspect.py
@@ -933,7 +933,7 @@ FullArgSpec = namedtuple('FullArgSpec',
     'args, varargs, varkw, defaults, kwonlyargs, kwonlydefaults, annotations')
 
 def getfullargspec(func):
-    """Get the names and default values of a function's arguments.
+    """Get the names and default values of a callable object's arguments.
 
     A tuple of seven things is returned:
     (args, varargs, varkw, defaults, kwonlyargs, kwonlydefaults annotations).
@@ -949,10 +949,15 @@ def getfullargspec(func):
 
     if ismethod(func):
         func = func.__func__
-    if not isfunction(func):
-        raise TypeError('{!r} is not a Python function'.format(func))
 
-    sig = signature(func)
+    try:
+        sig = signature(func)
+    except Exception as ex:
+        # Most of the times 'signature' will raise ValueError.
+        # But, it can also raise AttributeError, and, maybe something
+        # else. So to be fully backwards compatible, we catch all
+        # exceptions here, and reraise a TypeError.
+        raise TypeError('unsupported callable') from ex
 
     args = []
     varargs = None
diff --git a/Lib/test/test_inspect.py b/Lib/test/test_inspect.py
index 8660949..aad072f 100644
--- a/Lib/test/test_inspect.py
+++ b/Lib/test/test_inspect.py
@@ -566,7 +566,7 @@ class TestClassesAndFunctions(unittest.TestCase):
                                      kwonlyargs_e=['arg'],
                                      formatted='(*, arg)')
 
-    def test_getfullargspec_signature(self):
+    def test_getfullargspec_signature_attr(self):
         def test():
             pass
         spam_param = inspect.Parameter('spam', inspect.Parameter.POSITIONAL_ONLY)
@@ -574,6 +574,18 @@ class TestClassesAndFunctions(unittest.TestCase):
 
         self.assertFullArgSpecEquals(test, args_e=['spam'], formatted='(spam)')
 
+    @unittest.skipIf(MISSING_C_DOCSTRINGS,
+                     "Signature information for builtins requires docstrings")
+    def test_getfullagrspec_builtin_func(self):
+        builtin = _testcapi.docstring_with_signature_with_defaults
+        spec = inspect.getfullargspec(builtin)
+        self.assertEqual(spec.defaults[0], 'avocado')
+
+    def test_getfullagrspec_builtin_func_no_signature(self):
+        builtin = _testcapi.docstring_no_signature
+        with self.assertRaises(TypeError):
+            inspect.getfullargspec(builtin)
+
     def test_getargspec_method(self):
         class A(object):
             def m(self):