diff --git a/cf_xarray/accessor.py b/cf_xarray/accessor.py
index 14d7314e..f0a84bdf 100644
--- a/cf_xarray/accessor.py
+++ b/cf_xarray/accessor.py
@@ -1317,7 +1317,7 @@ def rename_like(
             Variables will be renamed to match variable names in this xarray object
         skip: str, Iterable[str], optional
             Limit the renaming excluding
-            ("axes", "cell_measures", "coordinates", "standard_names")
+            ("axes", "bounds", cell_measures", "coordinates", "standard_names")
             or a subset thereof.
 
         Returns
@@ -1332,24 +1332,48 @@ def rename_like(
         good_keys = ourkeys & theirkeys
         keydict = {}
         for key in good_keys:
-            ours = set(_get_all(self._obj, key))
-            theirs = set(_get_all(other, key))
+            ours = set(apply_mapper(_get_all, self._obj, key))
+            theirs = set(apply_mapper(_get_all, other, key))
             for attr in skip:
-                ours -= set(getattr(self, attr).get(key, []))
-                theirs -= set(getattr(other.cf, attr).get(key, []))
+                ours.difference_update(getattr(self, attr).get(key, []))
+                theirs.difference_update(getattr(other.cf, attr).get(key, []))
             if ours and theirs:
                 keydict[key] = dict(ours=list(ours), theirs=list(theirs))
 
-        conflicts = {}
-        for k0, v0 in keydict.items():
-            if len(v0["ours"]) > 1 or len(v0["theirs"]) > 1:
-                conflicts[k0] = v0
-                continue
-            for v1 in keydict.values():
-                # Conflicts have same ours but different theirs or vice versa
-                if (v0["ours"] == v1["ours"]) != (v0["theirs"] == v1["theirs"]):
+        def get_renamer_and_conflicts(keydict):
+            conflicts = {}
+            for k0, v0 in keydict.items():
+                if len(v0["ours"]) > 1 or len(v0["theirs"]) > 1:
                     conflicts[k0] = v0
-                    break
+                    continue
+                for v1 in keydict.values():
+                    # Conflicts have same ours but different theirs or vice versa
+                    if (v0["ours"] == v1["ours"]) != (v0["theirs"] == v1["theirs"]):
+                        conflicts[k0] = v0
+                        break
+
+            renamer = {
+                v["ours"][0]: v["theirs"][0]
+                for k, v in keydict.items()
+                if k not in conflicts
+            }
+
+            return renamer, conflicts
+
+        # Run get_renamer_and_conflicts twice.
+        # The second time add the bounds associated with variables to rename
+        renamer, conflicts = get_renamer_and_conflicts(keydict)
+        if "bounds" not in skip:
+            for k, v in renamer.items():
+                ours = set(getattr(self, "bounds", {}).get(k, []))
+                theirs = set(getattr(other.cf, "bounds", {}).get(v, []))
+                if ours and theirs:
+                    ours.update(keydict.get(k, {}).get("ours", []))
+                    theirs.update(keydict.get(k, {}).get("theirs", []))
+                    keydict[k] = dict(ours=list(ours), theirs=list(theirs))
+            renamer, conflicts = get_renamer_and_conflicts(keydict)
+
+        # Rename and warn
         if conflicts:
             warnings.warn(
                 "Conflicting variables skipped:\n"
@@ -1363,23 +1387,28 @@ def rename_like(
                 ),
                 UserWarning,
             )
-
-        renamer = {
-            v["ours"][0]: v["theirs"][0]
-            for k, v in keydict.items()
-            if k not in conflicts
-        }
         newobj = self._obj.rename(renamer)
 
-        # rename variable names in the coordinates attribute
+        # rename variable names in the attributes
         # if present
         ds = self._maybe_to_dataset(newobj)
         for _, variable in ds.variables.items():
-            coordinates = variable.attrs.get("coordinates", None)
-            if coordinates:
-                for k, v in renamer.items():
-                    coordinates = coordinates.replace(k, v)
-                variable.attrs["coordinates"] = coordinates
+            for attr in ("bounds", "coordinates", "cell_measures"):
+                if attr == "cell_measures":
+                    varlist = [
+                        f"{k}: {renamer.get(v, v)}"
+                        for k, v in parse_cell_methods_attr(
+                            variable.attrs.get(attr, "")
+                        ).items()
+                    ]
+                else:
+                    varlist = [
+                        renamer.get(var, var)
+                        for var in variable.attrs.get(attr, "").split()
+                    ]
+
+                if varlist:
+                    variable.attrs[attr] = " ".join(varlist)
         return self._maybe_to_dataarray(ds)
 
     def guess_coord_axis(self, verbose: bool = False) -> Union[DataArray, Dataset]:
diff --git a/cf_xarray/tests/test_accessor.py b/cf_xarray/tests/test_accessor.py
index e89a4a25..f2e65640 100644
--- a/cf_xarray/tests/test_accessor.py
+++ b/cf_xarray/tests/test_accessor.py
@@ -241,7 +241,7 @@ def test_rename_like():
     for k in ["TLONG", "TLAT"]:
         assert k not in renamed.coords
         assert k in original.coords
-        assert original.TEMP.attrs["coordinates"] == "TLONG TLAT"
+    assert original.TEMP.attrs["coordinates"] == "TLONG TLAT"
 
     assert "lon" in renamed.coords
     assert "lat" in renamed.coords
@@ -271,6 +271,20 @@ def test_rename_like():
     actual = da.cf.rename_like(airds, skip="axes").cf.coordinates
     assert expected == actual
 
+    # rename bounds
+    original = airds.cf[["air"]].cf.add_bounds("lon")
+    other = popds.cf[["TEMP"]].cf.add_bounds("nlon")
+    renamed = original.cf.rename_like(other, skip="coordinates")
+    assert renamed.cf.bounds["nlon"] == ["nlon_bounds"]
+
+    # rename cell measures
+    other = airds.cf["air"].cf.rename(area="CELL_AREA")
+    other.attrs["cell_measures"] = other.attrs["cell_measures"].replace(
+        "cell_area", "CELL_AREA"
+    )
+    renamed = airds.cf["air"].cf.rename_like(other)
+    assert renamed.cf.cell_measures["area"] == ["CELL_AREA"]
+
 
 @pytest.mark.parametrize("obj", objects)
 @pytest.mark.parametrize(
diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index fb03011f..9f444669 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -7,6 +7,7 @@ v0.5.3 (unreleased)
 ===================
 - Begin adding support for units with a unit registry for pint arrays. :pr:`197`.
   By `Jon Thielen`_ and `Justus Magin`_.
+- :py:meth:`Dataset.cf.rename_like` also updates the ``bounds`` and ``cell_measures`` attributes. By `Mattia Almansi`_.
 
 v0.5.2 (May 11, 2021)
 =====================