diff --git a/doc/source/whatsnew/v0.21.0.txt b/doc/source/whatsnew/v0.21.0.txt
index 8b2c4d16f4e1a..05d35545593c0 100644
--- a/doc/source/whatsnew/v0.21.0.txt
+++ b/doc/source/whatsnew/v0.21.0.txt
@@ -369,6 +369,7 @@ Reshaping
 - Fixes regression from 0.20, :func:`Series.aggregate` and :func:`DataFrame.aggregate` allow dictionaries as return values again (:issue:`16741`)
 - Fixes dtype of result with integer dtype input, from :func:`pivot_table` when called with ``margins=True`` (:issue:`17013`)
 - Bug in :func:`crosstab` where passing two ``Series`` with the same name raised a ``KeyError`` (:issue:`13279`)
+- :func:`Series.argmin`, :func:`Series.argmax`, and their counterparts on ``DataFrame`` and groupby objects work correctly with floating point data that contains infinite values (:issue:`13595`).
 
 Numeric
 ^^^^^^^
diff --git a/pandas/core/nanops.py b/pandas/core/nanops.py
index 2f4e437c0ae61..b2bbf1c75b7ea 100644
--- a/pandas/core/nanops.py
+++ b/pandas/core/nanops.py
@@ -486,23 +486,23 @@ def reduction(values, axis=None, skipna=True):
 nanmax = _nanminmax('max', fill_value_typ='-inf')
 
 
+@disallow('O')
 def nanargmax(values, axis=None, skipna=True):
     """
     Returns -1 in the NA case
     """
-    values, mask, dtype, _ = _get_values(values, skipna, fill_value_typ='-inf',
-                                         isfinite=True)
+    values, mask, dtype, _ = _get_values(values, skipna, fill_value_typ='-inf')
     result = values.argmax(axis)
     result = _maybe_arg_null_out(result, axis, mask, skipna)
     return result
 
 
+@disallow('O')
 def nanargmin(values, axis=None, skipna=True):
     """
     Returns -1 in the NA case
     """
-    values, mask, dtype, _ = _get_values(values, skipna, fill_value_typ='+inf',
-                                         isfinite=True)
+    values, mask, dtype, _ = _get_values(values, skipna, fill_value_typ='+inf')
     result = values.argmin(axis)
     result = _maybe_arg_null_out(result, axis, mask, skipna)
     return result
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index 0dea1e8447b2b..9983adc95b68f 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -2339,7 +2339,8 @@ def test_non_cython_api(self):
         assert_frame_equal(result, expected)
 
         # idxmax
-        expected = DataFrame([[0], [nan]], columns=['B'], index=[1, 3])
+        expected = DataFrame([[0.0], [nan]], columns=['B'],
+                             index=[1, 3])
         expected.index.name = 'A'
         result = g.idxmax()
         assert_frame_equal(result, expected)
diff --git a/pandas/tests/series/test_operators.py b/pandas/tests/series/test_operators.py
index 991c5ff625554..9710e480188e3 100644
--- a/pandas/tests/series/test_operators.py
+++ b/pandas/tests/series/test_operators.py
@@ -1857,3 +1857,45 @@ def test_op_duplicate_index(self):
         result = s1 + s2
         expected = pd.Series([11, 12, np.nan], index=[1, 1, 2])
         assert_series_equal(result, expected)
+
+    def test_argminmax(self):
+        # Series.argmin, Series.argmax are aliased to Series.idxmin,
+        # Series.idxmax
+
+        # Expected behavior for empty Series
+        s = pd.Series([])
+
+        with pytest.raises(ValueError):
+            s.argmin()
+        with pytest.raises(ValueError):
+            s.argmin(skipna=False)
+        with pytest.raises(ValueError):
+            s.argmax()
+        with pytest.raises(ValueError):
+            s.argmax(skipna=False)
+
+        # For numeric data with NA and Inf (GH #13595)
+        s = pd.Series([0, -np.inf, np.inf, np.nan])
+
+        assert s.argmin() == 1
+        assert np.isnan(s.argmin(skipna=False))
+
+        assert s.argmax() == 2
+        assert np.isnan(s.argmax(skipna=False))
+
+        # Using old-style behavior that treats floating point nan, -inf, and
+        # +inf as missing
+        s = pd.Series([0, -np.inf, np.inf, np.nan])
+
+        with pd.option_context('mode.use_inf_as_null', True):
+            assert s.argmin() == 0
+            assert np.isnan(s.argmin(skipna=False))
+            assert s.argmax() == 0
+            np.isnan(s.argmax(skipna=False))
+
+        # For strings and mixed data types
+        s = pd.Series(['foo', 'foo', 'bar', 'bar', None, np.nan, 'baz'])
+        with pytest.raises(TypeError):
+            s.argmin()
+        with pytest.raises(TypeError):
+            s.argmax()