diff --git a/doc/source/whatsnew/v0.21.0.txt b/doc/source/whatsnew/v0.21.0.txt
index 935e9d740b91c..b8c4cf61edc5b 100644
--- a/doc/source/whatsnew/v0.21.0.txt
+++ b/doc/source/whatsnew/v0.21.0.txt
@@ -145,6 +145,7 @@ Bug Fixes
 ~~~~~~~~~
 
 - Fixes regression in 0.20, :func:`Series.aggregate` and :func:`DataFrame.aggregate` allow dictionaries as return values again (:issue:`16741`)
+- Fixes bug where indexing with ``np.inf`` caused an ``OverflowError`` to be raised (:issue:`16957`)
 
 Conversion
 ^^^^^^^^^^
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index bbbc19b36964d..5d50f961927c7 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -666,7 +666,7 @@ def _try_convert_to_int_index(cls, data, copy, name):
             res = data.astype('u8', copy=False)
             if (res == data).all():
                 return UInt64Index(res, copy=copy, name=name)
-        except (TypeError, ValueError):
+        except (OverflowError, TypeError, ValueError):
             pass
 
         raise ValueError
@@ -1640,7 +1640,7 @@ def __contains__(self, key):
         hash(key)
         try:
             return key in self._engine
-        except (TypeError, ValueError):
+        except (OverflowError, TypeError, ValueError):
             return False
 
     _index_shared_docs['contains'] = """
@@ -3365,7 +3365,7 @@ def _maybe_cast_indexer(self, key):
                 ckey = int(key)
                 if ckey == key:
                     key = ckey
-            except (ValueError, TypeError):
+            except (OverflowError, ValueError, TypeError):
                 pass
         return key
 
diff --git a/pandas/tests/indexing/test_indexing.py b/pandas/tests/indexing/test_indexing.py
index 9fa677eb624ae..98f5d5eb140df 100644
--- a/pandas/tests/indexing/test_indexing.py
+++ b/pandas/tests/indexing/test_indexing.py
@@ -63,6 +63,34 @@ def f():
 
         pytest.raises(ValueError, f)
 
+    def test_inf_upcast(self):
+        # GH 16957
+        # We should be able to use np.inf as a key
+        # np.inf should cause an index to convert to float
+
+        # Test with np.inf in rows
+        df = pd.DataFrame(columns=[0])
+        df.loc[1] = 1
+        df.loc[2] = 2
+        df.loc[np.inf] = 3
+
+        # make sure we can look up the value
+        assert df.loc[np.inf, 0] == 3
+
+        result = df.index
+        expected = pd.Float64Index([1, 2, np.inf])
+        tm.assert_index_equal(result, expected)
+
+        # Test with np.inf in columns
+        df = pd.DataFrame()
+        df.loc[0, 0] = 1
+        df.loc[1, 1] = 2
+        df.loc[0, np.inf] = 3
+
+        result = df.columns
+        expected = pd.Float64Index([0, 1, np.inf])
+        tm.assert_index_equal(result, expected)
+
     def test_setitem_dtype_upcast(self):
 
         # GH3216
@@ -542,6 +570,34 @@ def test_astype_assignment_with_dups(self):
         # result = df.get_dtype_counts().sort_index()
         # expected = Series({'float64': 2, 'object': 1}).sort_index()
 
+    @pytest.mark.parametrize("index,val", [
+        (pd.Index([0, 1, 2]), 2),
+        (pd.Index([0, 1, '2']), '2'),
+        (pd.Index([0, 1, 2, np.inf, 4]), 4),
+        (pd.Index([0, 1, 2, np.nan, 4]), 4),
+        (pd.Index([0, 1, 2, np.inf]), np.inf),
+        (pd.Index([0, 1, 2, np.nan]), np.nan),
+    ])
+    def test_index_contains(self, index, val):
+        assert val in index
+
+    @pytest.mark.parametrize("index,val", [
+        (pd.Index([0, 1, 2]), '2'),
+        (pd.Index([0, 1, '2']), 2),
+        (pd.Index([0, 1, 2, np.inf]), 4),
+        (pd.Index([0, 1, 2, np.nan]), 4),
+        (pd.Index([0, 1, 2, np.inf]), np.nan),
+        (pd.Index([0, 1, 2, np.nan]), np.inf),
+        # Checking if np.inf in Int64Index should not cause an OverflowError
+        # Related to GH 16957
+        (pd.Int64Index([0, 1, 2]), np.inf),
+        (pd.Int64Index([0, 1, 2]), np.nan),
+        (pd.UInt64Index([0, 1, 2]), np.inf),
+        (pd.UInt64Index([0, 1, 2]), np.nan),
+    ])
+    def test_index_not_contains(self, index, val):
+        assert val not in index
+
     def test_index_type_coercion(self):
 
         with catch_warnings(record=True):