diff --git a/doc/api-hidden.rst b/doc/api-hidden.rst
index 076b0eb452a..fc27d9c3fe8 100644
--- a/doc/api-hidden.rst
+++ b/doc/api-hidden.rst
@@ -54,7 +54,6 @@
    core.rolling.DatasetCoarsen.var
    core.rolling.DatasetCoarsen.boundary
    core.rolling.DatasetCoarsen.coord_func
-   core.rolling.DatasetCoarsen.keep_attrs
    core.rolling.DatasetCoarsen.obj
    core.rolling.DatasetCoarsen.side
    core.rolling.DatasetCoarsen.trim_excess
@@ -120,7 +119,6 @@
    core.rolling.DatasetRolling.var
    core.rolling.DatasetRolling.center
    core.rolling.DatasetRolling.dim
-   core.rolling.DatasetRolling.keep_attrs
    core.rolling.DatasetRolling.min_periods
    core.rolling.DatasetRolling.obj
    core.rolling.DatasetRolling.rollings
@@ -199,7 +197,6 @@
    core.rolling.DataArrayCoarsen.var
    core.rolling.DataArrayCoarsen.boundary
    core.rolling.DataArrayCoarsen.coord_func
-   core.rolling.DataArrayCoarsen.keep_attrs
    core.rolling.DataArrayCoarsen.obj
    core.rolling.DataArrayCoarsen.side
    core.rolling.DataArrayCoarsen.trim_excess
@@ -263,7 +260,6 @@
    core.rolling.DataArrayRolling.var
    core.rolling.DataArrayRolling.center
    core.rolling.DataArrayRolling.dim
-   core.rolling.DataArrayRolling.keep_attrs
    core.rolling.DataArrayRolling.min_periods
    core.rolling.DataArrayRolling.obj
    core.rolling.DataArrayRolling.window
diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 8d8598b6830..6127edcfc87 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -77,6 +77,10 @@ Breaking changes
 Deprecations
 ~~~~~~~~~~~~
 
+- Removed the deprecated ``dim`` kwarg to :py:func:`DataArray.integrate` (:pull:`5630`)
+- Removed the deprecated ``keep_attrs`` kwarg to :py:func:`DataArray.rolling` (:pull:`5630`)
+- Removed the deprecated ``keep_attrs`` kwarg to :py:func:`DataArray.coarsen` (:pull:`5630`)
+- Completed deprecation of passing an ``xarray.DataArray`` to :py:func:`Variable` - will now raise a ``TypeError`` (:pull:`5630`)
 
 Bug fixes
 ~~~~~~~~~
diff --git a/xarray/core/combine.py b/xarray/core/combine.py
index de6d16ef5c3..be9c2992832 100644
--- a/xarray/core/combine.py
+++ b/xarray/core/combine.py
@@ -635,7 +635,7 @@ def _combine_single_variable_hypercube(
     return concatenated
 
 
-# TODO remove empty list default param after version 0.19, see PR4696
+# TODO remove empty list default param after version 0.21, see PR4696
 def combine_by_coords(
     data_objects=[],
     compat="no_conflicts",
@@ -849,11 +849,11 @@ def combine_by_coords(
         precipitation  (y, x) float64 0.4376 0.8918 0.9637 ... 0.5684 0.01879 0.6176
     """
 
-    # TODO remove after version 0.19, see PR4696
+    # TODO remove after version 0.21, see PR4696
     if datasets is not None:
         warnings.warn(
             "The datasets argument has been renamed to `data_objects`."
-            " In future passing a value for datasets will raise an error."
+            " From 0.21 on passing a value for datasets will raise an error."
         )
         data_objects = datasets
 
diff --git a/xarray/core/common.py b/xarray/core/common.py
index 7b6e9198b43..ab822f576d3 100644
--- a/xarray/core/common.py
+++ b/xarray/core/common.py
@@ -821,7 +821,6 @@ def rolling(
         dim: Mapping[Hashable, int] = None,
         min_periods: int = None,
         center: Union[bool, Mapping[Hashable, bool]] = False,
-        keep_attrs: bool = None,
         **window_kwargs: int,
     ):
         """
@@ -889,9 +888,7 @@ def rolling(
         """
 
         dim = either_dict_or_kwargs(dim, window_kwargs, "rolling")
-        return self._rolling_cls(
-            self, dim, min_periods=min_periods, center=center, keep_attrs=keep_attrs
-        )
+        return self._rolling_cls(self, dim, min_periods=min_periods, center=center)
 
     def rolling_exp(
         self,
@@ -940,7 +937,6 @@ def coarsen(
         boundary: str = "exact",
         side: Union[str, Mapping[Hashable, str]] = "left",
         coord_func: str = "mean",
-        keep_attrs: bool = None,
         **window_kwargs: int,
     ):
         """
@@ -1009,7 +1005,6 @@ def coarsen(
             boundary=boundary,
             side=side,
             coord_func=coord_func,
-            keep_attrs=keep_attrs,
         )
 
     def resample(
diff --git a/xarray/core/dataarray.py b/xarray/core/dataarray.py
index 46c488b83a9..cb2c4d30a69 100644
--- a/xarray/core/dataarray.py
+++ b/xarray/core/dataarray.py
@@ -3582,8 +3582,6 @@ def integrate(
         self,
         coord: Union[Hashable, Sequence[Hashable]] = None,
         datetime_unit: str = None,
-        *,
-        dim: Union[Hashable, Sequence[Hashable]] = None,
     ) -> "DataArray":
         """Integrate along the given coordinate using the trapezoidal rule.
 
@@ -3595,8 +3593,6 @@ def integrate(
         ----------
         coord : hashable, or sequence of hashable
             Coordinate(s) used for the integration.
-        dim : hashable, or sequence of hashable
-            Coordinate(s) used for the integration.
         datetime_unit : {'Y', 'M', 'W', 'D', 'h', 'm', 's', 'ms', 'us', 'ns', \
                         'ps', 'fs', 'as'}, optional
             Specify the unit if a datetime coordinate is used.
@@ -3633,21 +3629,6 @@ def integrate(
         array([5.4, 6.6, 7.8])
         Dimensions without coordinates: y
         """
-        if dim is not None and coord is not None:
-            raise ValueError(
-                "Cannot pass both 'dim' and 'coord'. Please pass only 'coord' instead."
-            )
-
-        if dim is not None and coord is None:
-            coord = dim
-            msg = (
-                "The `dim` keyword argument to `DataArray.integrate` is "
-                "being replaced with `coord`, for consistency with "
-                "`Dataset.integrate`. Please pass `coord` instead."
-                " `dim` will be removed in version 0.19.0."
-            )
-            warnings.warn(msg, FutureWarning, stacklevel=2)
-
         ds = self._to_temp_dataset().integrate(coord, datetime_unit)
         return self._from_temp_dataset(ds)
 
diff --git a/xarray/core/dataset.py b/xarray/core/dataset.py
index 650b3c80adf..5f5c01ad4c9 100644
--- a/xarray/core/dataset.py
+++ b/xarray/core/dataset.py
@@ -4173,6 +4173,7 @@ def update(self, other: "CoercibleMapping") -> "Dataset":
         """Update this dataset's variables with those from another dataset.
 
         Just like :py:meth:`dict.update` this is a in-place operation.
+        For a non-inplace version, see :py:meth:`Dataset.merge`.
 
         Parameters
         ----------
@@ -4191,7 +4192,7 @@ def update(self, other: "CoercibleMapping") -> "Dataset":
             Updated dataset. Note that since the update is in-place this is the input
             dataset.
 
-            It is deprecated since version 0.17 and scheduled to be removed in 0.19.
+            It is deprecated since version 0.17 and scheduled to be removed in 0.21.
 
         Raises
         ------
@@ -4202,6 +4203,7 @@ def update(self, other: "CoercibleMapping") -> "Dataset":
         See Also
         --------
         Dataset.assign
+        Dataset.merge
         """
         merge_result = dataset_update_method(self, other)
         return self._replace(inplace=True, **merge_result._asdict())
@@ -4275,6 +4277,10 @@ def merge(
         ------
         MergeError
             If any variables conflict (see ``compat``).
+
+        See Also
+        --------
+        Dataset.update
         """
         other = other.to_dataset() if isinstance(other, xr.DataArray) else other
         merge_result = dataset_merge_method(
diff --git a/xarray/core/rolling.py b/xarray/core/rolling.py
index b87dcda24b0..04052510f5d 100644
--- a/xarray/core/rolling.py
+++ b/xarray/core/rolling.py
@@ -48,10 +48,10 @@ class Rolling:
     xarray.DataArray.rolling
     """
 
-    __slots__ = ("obj", "window", "min_periods", "center", "dim", "keep_attrs")
-    _attributes = ("window", "min_periods", "center", "dim", "keep_attrs")
+    __slots__ = ("obj", "window", "min_periods", "center", "dim")
+    _attributes = ("window", "min_periods", "center", "dim")
 
-    def __init__(self, obj, windows, min_periods=None, center=False, keep_attrs=None):
+    def __init__(self, obj, windows, min_periods=None, center=False):
         """
         Moving window object.
 
@@ -89,15 +89,6 @@ def __init__(self, obj, windows, min_periods=None, center=False, keep_attrs=None
 
         self.min_periods = np.prod(self.window) if min_periods is None else min_periods
 
-        if keep_attrs is not None:
-            warnings.warn(
-                "Passing ``keep_attrs`` to ``rolling`` is deprecated and will raise an"
-                " error in xarray 0.18. Please pass ``keep_attrs`` directly to the"
-                " applied function. Note that keep_attrs is now True per default.",
-                FutureWarning,
-            )
-        self.keep_attrs = keep_attrs
-
     def __repr__(self):
         """provide a nice str repr of our rolling object"""
 
@@ -188,15 +179,8 @@ def _mapping_to_list(
             )
 
     def _get_keep_attrs(self, keep_attrs):
-
         if keep_attrs is None:
-            # TODO: uncomment the next line and remove the others after the deprecation
-            # keep_attrs = _get_keep_attrs(default=True)
-
-            if self.keep_attrs is None:
-                keep_attrs = _get_keep_attrs(default=True)
-            else:
-                keep_attrs = self.keep_attrs
+            keep_attrs = _get_keep_attrs(default=True)
 
         return keep_attrs
 
@@ -204,7 +188,7 @@ def _get_keep_attrs(self, keep_attrs):
 class DataArrayRolling(Rolling):
     __slots__ = ("window_labels",)
 
-    def __init__(self, obj, windows, min_periods=None, center=False, keep_attrs=None):
+    def __init__(self, obj, windows, min_periods=None, center=False):
         """
         Moving window object for DataArray.
         You should use DataArray.rolling() method to construct this object
@@ -235,9 +219,7 @@ def __init__(self, obj, windows, min_periods=None, center=False, keep_attrs=None
         xarray.Dataset.rolling
         xarray.Dataset.groupby
         """
-        super().__init__(
-            obj, windows, min_periods=min_periods, center=center, keep_attrs=keep_attrs
-        )
+        super().__init__(obj, windows, min_periods=min_periods, center=center)
 
         # TODO legacy attribute
         self.window_labels = self.obj[self.dim[0]]
@@ -561,7 +543,7 @@ def _numpy_or_bottleneck_reduce(
 class DatasetRolling(Rolling):
     __slots__ = ("rollings",)
 
-    def __init__(self, obj, windows, min_periods=None, center=False, keep_attrs=None):
+    def __init__(self, obj, windows, min_periods=None, center=False):
         """
         Moving window object for Dataset.
         You should use Dataset.rolling() method to construct this object
@@ -592,7 +574,7 @@ def __init__(self, obj, windows, min_periods=None, center=False, keep_attrs=None
         xarray.Dataset.groupby
         xarray.DataArray.groupby
         """
-        super().__init__(obj, windows, min_periods, center, keep_attrs)
+        super().__init__(obj, windows, min_periods, center)
         if any(d not in self.obj.dims for d in self.dim):
             raise KeyError(self.dim)
         # Keep each Rolling object as a dictionary
@@ -768,11 +750,10 @@ class Coarsen(CoarsenArithmetic):
         "windows",
         "side",
         "trim_excess",
-        "keep_attrs",
     )
     _attributes = ("windows", "side", "trim_excess")
 
-    def __init__(self, obj, windows, boundary, side, coord_func, keep_attrs):
+    def __init__(self, obj, windows, boundary, side, coord_func):
         """
         Moving window object.
 
@@ -799,17 +780,6 @@ def __init__(self, obj, windows, boundary, side, coord_func, keep_attrs):
         self.side = side
         self.boundary = boundary
 
-        if keep_attrs is not None:
-            warnings.warn(
-                "Passing ``keep_attrs`` to ``coarsen`` is deprecated and will raise an"
-                " error in xarray 0.19. Please pass ``keep_attrs`` directly to the"
-                " applied function, i.e. use ``ds.coarsen(...).mean(keep_attrs=False)``"
-                " instead of ``ds.coarsen(..., keep_attrs=False).mean()``"
-                " Note that keep_attrs is now True per default.",
-                FutureWarning,
-            )
-        self.keep_attrs = keep_attrs
-
         absent_dims = [dim for dim in windows.keys() if dim not in self.obj.dims]
         if absent_dims:
             raise ValueError(
@@ -823,15 +793,8 @@ def __init__(self, obj, windows, boundary, side, coord_func, keep_attrs):
         self.coord_func = coord_func
 
     def _get_keep_attrs(self, keep_attrs):
-
         if keep_attrs is None:
-            # TODO: uncomment the next line and remove the others after the deprecation
-            # keep_attrs = _get_keep_attrs(default=True)
-
-            if self.keep_attrs is None:
-                keep_attrs = _get_keep_attrs(default=True)
-            else:
-                keep_attrs = self.keep_attrs
+            keep_attrs = _get_keep_attrs(default=True)
 
         return keep_attrs
 
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index f228ef43e32..f69951580c7 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -118,14 +118,9 @@ def as_variable(obj, name=None) -> "Union[Variable, IndexVariable]":
         obj = obj.copy(deep=False)
     elif isinstance(obj, tuple):
         if isinstance(obj[1], DataArray):
-            # TODO: change into TypeError
-            warnings.warn(
-                (
-                    "Using a DataArray object to construct a variable is"
-                    " ambiguous, please extract the data using the .data property."
-                    " This will raise a TypeError in 0.19.0."
-                ),
-                DeprecationWarning,
+            raise TypeError(
+                "Using a DataArray object to construct a variable is"
+                " ambiguous, please extract the data using the .data property."
             )
         try:
             obj = Variable(*obj)
diff --git a/xarray/tests/test_coarsen.py b/xarray/tests/test_coarsen.py
index 503c742252a..278a961166f 100644
--- a/xarray/tests/test_coarsen.py
+++ b/xarray/tests/test_coarsen.py
@@ -153,39 +153,6 @@ def test_coarsen_keep_attrs(funcname, argument):
     assert result.da_not_coarsend.name == "da_not_coarsend"
 
 
-def test_coarsen_keep_attrs_deprecated():
-    global_attrs = {"units": "test", "long_name": "testing"}
-    attrs_da = {"da_attr": "test"}
-
-    data = np.linspace(10, 15, 100)
-    coords = np.linspace(1, 10, 100)
-
-    ds = Dataset(
-        data_vars={"da": ("coord", data)},
-        coords={"coord": coords},
-        attrs=global_attrs,
-    )
-    ds.da.attrs = attrs_da
-
-    # deprecated option
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``coarsen`` is deprecated"
-    ):
-        result = ds.coarsen(dim={"coord": 5}, keep_attrs=False).mean()
-
-    assert result.attrs == {}
-    assert result.da.attrs == {}
-
-    # the keep_attrs in the reduction function takes precedence
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``coarsen`` is deprecated"
-    ):
-        result = ds.coarsen(dim={"coord": 5}, keep_attrs=True).mean(keep_attrs=False)
-
-    assert result.attrs == {}
-    assert result.da.attrs == {}
-
-
 @pytest.mark.slow
 @pytest.mark.parametrize("ds", (1, 2), indirect=True)
 @pytest.mark.parametrize("window", (1, 2, 3, 4))
@@ -267,31 +234,6 @@ def test_coarsen_da_keep_attrs(funcname, argument):
     assert result.name == "name"
 
 
-def test_coarsen_da_keep_attrs_deprecated():
-    attrs_da = {"da_attr": "test"}
-
-    data = np.linspace(10, 15, 100)
-    coords = np.linspace(1, 10, 100)
-
-    da = DataArray(data, dims=("coord"), coords={"coord": coords}, attrs=attrs_da)
-
-    # deprecated option
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``coarsen`` is deprecated"
-    ):
-        result = da.coarsen(dim={"coord": 5}, keep_attrs=False).mean()
-
-    assert result.attrs == {}
-
-    # the keep_attrs in the reduction function takes precedence
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``coarsen`` is deprecated"
-    ):
-        result = da.coarsen(dim={"coord": 5}, keep_attrs=True).mean(keep_attrs=False)
-
-    assert result.attrs == {}
-
-
 @pytest.mark.parametrize("da", (1, 2), indirect=True)
 @pytest.mark.parametrize("window", (1, 2, 3, 4))
 @pytest.mark.parametrize("name", ("sum", "mean", "std", "max"))
diff --git a/xarray/tests/test_dask.py b/xarray/tests/test_dask.py
index 0a4b8725be6..d5d460056aa 100644
--- a/xarray/tests/test_dask.py
+++ b/xarray/tests/test_dask.py
@@ -1593,7 +1593,7 @@ def test_more_transforms_pass_lazy_array_equiv(map_da, map_ds):
         assert_equal(xr.broadcast(map_ds.cxy, map_ds.cxy)[0], map_ds.cxy)
         assert_equal(map_ds.map(lambda x: x), map_ds)
         assert_equal(map_ds.set_coords("a").reset_coords("a"), map_ds)
-        assert_equal(map_ds.update({"a": map_ds.a}), map_ds)
+        assert_equal(map_ds.assign({"a": map_ds.a}), map_ds)
 
         # fails because of index error
         # assert_equal(
diff --git a/xarray/tests/test_dataarray.py b/xarray/tests/test_dataarray.py
index 79370adb016..012b070f1ee 100644
--- a/xarray/tests/test_dataarray.py
+++ b/xarray/tests/test_dataarray.py
@@ -6867,33 +6867,6 @@ def test_rolling_keep_attrs(funcname, argument):
     assert result.name == "name"
 
 
-def test_rolling_keep_attrs_deprecated():
-    attrs_da = {"da_attr": "test"}
-
-    data = np.linspace(10, 15, 100)
-    coords = np.linspace(1, 10, 100)
-
-    da = DataArray(data, dims=("coord"), coords={"coord": coords}, attrs=attrs_da)
-
-    # deprecated option
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``rolling`` is deprecated"
-    ):
-        result = da.rolling(dim={"coord": 5}, keep_attrs=False).construct("window_dim")
-
-    assert result.attrs == {}
-
-    # the keep_attrs in the reduction function takes precedence
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``rolling`` is deprecated"
-    ):
-        result = da.rolling(dim={"coord": 5}, keep_attrs=True).construct(
-            "window_dim", keep_attrs=False
-        )
-
-    assert result.attrs == {}
-
-
 def test_raise_no_warning_for_nan_in_binary_ops():
     with pytest.warns(None) as record:
         xr.DataArray([1, 2, np.NaN]) > 0
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index f6fb4aa0cde..02d27ade161 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -3193,13 +3193,13 @@ def test_update(self):
         data = create_test_data(seed=0)
         expected = data.copy()
         var2 = Variable("dim1", np.arange(8))
-        actual = data.update({"var2": var2})
+        actual = data
+        actual.update({"var2": var2})
         expected["var2"] = var2
         assert_identical(expected, actual)
 
         actual = data.copy()
-        actual_result = actual.update(data)
-        assert actual_result is actual
+        actual.update(data)
         assert_identical(expected, actual)
 
         other = Dataset(attrs={"new": "attr"})
@@ -6111,41 +6111,6 @@ def test_rolling_keep_attrs(funcname, argument):
     assert result.da_not_rolled.name == "da_not_rolled"
 
 
-def test_rolling_keep_attrs_deprecated():
-    global_attrs = {"units": "test", "long_name": "testing"}
-    attrs_da = {"da_attr": "test"}
-
-    data = np.linspace(10, 15, 100)
-    coords = np.linspace(1, 10, 100)
-
-    ds = Dataset(
-        data_vars={"da": ("coord", data)},
-        coords={"coord": coords},
-        attrs=global_attrs,
-    )
-    ds.da.attrs = attrs_da
-
-    # deprecated option
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``rolling`` is deprecated"
-    ):
-        result = ds.rolling(dim={"coord": 5}, keep_attrs=False).construct("window_dim")
-
-    assert result.attrs == {}
-    assert result.da.attrs == {}
-
-    # the keep_attrs in the reduction function takes precedence
-    with pytest.warns(
-        FutureWarning, match="Passing ``keep_attrs`` to ``rolling`` is deprecated"
-    ):
-        result = ds.rolling(dim={"coord": 5}, keep_attrs=True).construct(
-            "window_dim", keep_attrs=False
-        )
-
-    assert result.attrs == {}
-    assert result.da.attrs == {}
-
-
 def test_rolling_properties(ds):
     # catching invalid args
     with pytest.raises(ValueError, match="window must be > 0"):
@@ -6591,9 +6556,6 @@ def test_integrate(dask):
     with pytest.raises(ValueError):
         da.integrate("x2d")
 
-    with pytest.warns(FutureWarning):
-        da.integrate(dim="x")
-
 
 @requires_scipy
 @pytest.mark.parametrize("dask", [True, False])
diff --git a/xarray/tests/test_variable.py b/xarray/tests/test_variable.py
index d999bc1eb8b..96072a4e1e0 100644
--- a/xarray/tests/test_variable.py
+++ b/xarray/tests/test_variable.py
@@ -1162,7 +1162,7 @@ def test_as_variable(self):
         td = np.array([timedelta(days=x) for x in range(10)])
         assert as_variable(td, "time").dtype.kind == "m"
 
-        with pytest.warns(DeprecationWarning):
+        with pytest.raises(TypeError):
             as_variable(("x", DataArray([])))
 
     def test_repr(self):