diff --git a/cf_xarray/accessor.py b/cf_xarray/accessor.py
index ed8dea3e..6de5b08a 100644
--- a/cf_xarray/accessor.py
+++ b/cf_xarray/accessor.py
@@ -2,7 +2,7 @@
 import inspect
 import itertools
 import warnings
-from collections import ChainMap
+from collections import ChainMap, Counter
 from typing import (
     Any,
     Callable,
@@ -1298,8 +1298,6 @@ def rename_like(
         ``other``, that variable will be renamed to match the "longitude" variable in
         ``other``.
 
-        For now, this function only matches ``("latitude", "longitude", "vertical", "time")``
-
         Parameters
         ----------
         other: DataArray, Dataset
@@ -1312,18 +1310,21 @@ def rename_like(
         ourkeys = self.keys()
         theirkeys = other.cf.keys()
 
-        good_keys = set(_COORD_NAMES) & ourkeys & theirkeys
-        if not good_keys:
-            raise ValueError(
-                "No common coordinate variables between these two objects."
-            )
-
+        good_keys = ourkeys & theirkeys
         renamer = {}
         for key in good_keys:
-            ours = _get_axis_coord_single(self._obj, key)[0]
-            theirs = _get_axis_coord_single(other, key)[0]
-            renamer[ours] = theirs
+            ours = apply_mapper(
+                (_get_axis_coord, _get_with_standard_name), self._obj, key, error=False
+            )
+            theirs = apply_mapper(
+                (_get_axis_coord, _get_with_standard_name), other, key, error=False
+            )
+            if len(ours) > 1 or len(theirs) > 1:
+                continue
+            renamer[ours[0]] = theirs[0]
 
+        count = Counter(renamer.values())
+        renamer = {k: v for k, v in renamer.items() if count[v] == 1}
         newobj = self._obj.rename(renamer)
 
         # rename variable names in the coordinates attribute
diff --git a/cf_xarray/datasets.py b/cf_xarray/datasets.py
index 4e9e6c7b..5e6df446 100644
--- a/cf_xarray/datasets.py
+++ b/cf_xarray/datasets.py
@@ -122,7 +122,7 @@
 romsds["temp"] = (
     ("ocean_time", "s_rho"),
     [np.linspace(20, 30, 30)] * 2,
-    {"coordinates": "z_rho_dummy"},
+    {"coordinates": "z_rho_dummy", "standard_name": "sea_water_potential_temperature"},
 )
 romsds["temp"].encoding["coordinates"] = "s_rho"
 romsds.coords["z_rho_dummy"] = (
diff --git a/cf_xarray/tests/test_accessor.py b/cf_xarray/tests/test_accessor.py
index 138c6290..91fc353c 100644
--- a/cf_xarray/tests/test_accessor.py
+++ b/cf_xarray/tests/test_accessor.py
@@ -214,10 +214,13 @@ def test_getitem_ancillary_variables():
 def test_rename_like():
     original = popds.copy(deep=True)
 
-    with pytest.raises(KeyError):
-        popds.cf.rename_like(airds)
+    # it'll match for axis: X (lon, nlon) and coordinate="longitude" (lon, TLONG)
+    # so delete the axis attributes
+    newair = airds.copy(deep=True)
+    del newair.lon.attrs["axis"]
+    del newair.lat.attrs["axis"]
 
-    renamed = popds.cf["TEMP"].cf.rename_like(airds)
+    renamed = popds.cf["TEMP"].cf.rename_like(newair)
     for k in ["TLONG", "TLAT"]:
         assert k not in renamed.coords
         assert k in original.coords
@@ -227,6 +230,20 @@ def test_rename_like():
     assert "lat" in renamed.coords
     assert renamed.attrs["coordinates"] == "lon lat"
 
+    # standard name matching
+    newroms = romsds.expand_dims(latitude=[1], longitude=[1]).cf.guess_coord_axis()
+    renamed = popds.cf["UVEL"].cf.rename_like(newroms)
+    assert renamed.attrs["coordinates"] == "longitude latitude"
+    assert "longitude" in renamed.coords
+    assert "latitude" in renamed.coords
+    assert "ULON" not in renamed.coords
+    assert "ULAT" not in renamed.coords
+
+    # should change "temp" to "TEMP"
+    renamed = romsds.cf.rename_like(popds)
+    assert "temp" not in renamed
+    assert "TEMP" in renamed
+
 
 @pytest.mark.parametrize("obj", objects)
 @pytest.mark.parametrize(
diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index e6be6541..7d900984 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -10,7 +10,7 @@ v0.4.1 (unreleased)
 - Automatically set ``x`` or ``y`` for :py:attr:`DataArray.cf.plot`. By `Deepak Cherian`_.
 - Added scripts to document :ref:`criteria` with tables. By `Mattia Almansi`_.
 - Support for ``.drop()``, ``.drop_vars()``, ``.drop_sel()``, ``.drop_dims()``, ``.set_coords()``, ``.reset_coords()``. By `Mattia Almansi`_.
-- Support for using ``standard_name`` in more functions. (:pr:`128`) By `Deepak Cherian`_
+- Support for using ``standard_name`` in more functions. (:pr:`128`, :pr:`165`) By `Deepak Cherian`_
 - Allow :py:meth:`DataArray.cf.__getitem__` with standard names. By `Deepak Cherian`_
 - Rewrite the ``values`` of :py:attr:`Dataset.coords` and :py:attr:`Dataset.data_vars` with objects returned
   by :py:meth:`Dataset.cf.__getitem__`. This allows extraction of DataArrays when there are clashes