diff --git a/asv_bench/benchmarks/boolean.py b/asv_bench/benchmarks/boolean.py
new file mode 100644
index 0000000000000..71c422c641775
--- /dev/null
+++ b/asv_bench/benchmarks/boolean.py
@@ -0,0 +1,32 @@
+import numpy as np
+
+import pandas as pd
+
+
+class TimeLogicalOps:
+    def setup(self):
+        N = 10_000
+        left, right, lmask, rmask = np.random.randint(0, 2, size=(4, N)).astype("bool")
+        self.left = pd.arrays.BooleanArray(left, lmask)
+        self.right = pd.arrays.BooleanArray(right, rmask)
+
+    def time_or_scalar(self):
+        self.left | True
+        self.left | False
+
+    def time_or_array(self):
+        self.left | self.right
+
+    def time_and_scalar(self):
+        self.left & True
+        self.left & False
+
+    def time_and_array(self):
+        self.left & self.right
+
+    def time_xor_scalar(self):
+        self.left ^ True
+        self.left ^ False
+
+    def time_xor_array(self):
+        self.left ^ self.right
diff --git a/doc/source/index.rst.template b/doc/source/index.rst.template
index 9ec330c956ff1..9cea68530fbe7 100644
--- a/doc/source/index.rst.template
+++ b/doc/source/index.rst.template
@@ -73,6 +73,7 @@ See the :ref:`overview` for more detail about what's in the library.
   * :doc:`user_guide/missing_data`
   * :doc:`user_guide/categorical`
   * :doc:`user_guide/integer_na`
+  * :doc:`user_guide/boolean`
   * :doc:`user_guide/visualization`
   * :doc:`user_guide/computation`
   * :doc:`user_guide/groupby`
diff --git a/doc/source/user_guide/boolean.rst b/doc/source/user_guide/boolean.rst
new file mode 100644
index 0000000000000..e0f676d3072fc
--- /dev/null
+++ b/doc/source/user_guide/boolean.rst
@@ -0,0 +1,79 @@
+.. currentmodule:: pandas
+
+.. ipython:: python
+   :suppress:
+
+   import pandas as pd
+   import numpy as np
+
+.. _boolean:
+
+**************************
+Nullable Boolean Data Type
+**************************
+
+.. versionadded:: 1.0.0
+
+.. _boolean.kleene:
+
+Kleene Logical Operations
+-------------------------
+
+:class:`arrays.BooleanArray` implements `Kleene Logic`_ (sometimes called three-value logic) for
+logical operations like ``&`` (and), ``|`` (or) and ``^`` (exclusive-or).
+
+This table demonstrates the results for every combination. These operations are symmetrical,
+so flipping the left- and right-hand side makes no difference in the result.
+
+================= =========
+Expression        Result
+================= =========
+``True & True``   ``True``
+``True & False``  ``False``
+``True & NA``     ``NA``
+``False & False`` ``False``
+``False & NA``    ``False``
+``NA & NA``       ``NA``
+``True | True``   ``True``
+``True | False``  ``True``
+``True | NA``     ``True``
+``False | False`` ``False``
+``False | NA``    ``NA``
+``NA | NA``       ``NA``
+``True ^ True``   ``False``
+``True ^ False``  ``True``
+``True ^ NA``     ``NA``
+``False ^ False`` ``False``
+``False ^ NA``    ``NA``
+``NA ^ NA``       ``NA``
+================= =========
+
+When an ``NA`` is present in an operation, the output value is ``NA`` only if
+the result cannot be determined solely based on the other input. For example,
+``True | NA`` is ``True``, because both ``True | True`` and ``True | False``
+are ``True``. In that case, we don't actually need to consider the value
+of the ``NA``.
+
+On the other hand, ``True & NA`` is ``NA``. The result depends on whether
+the ``NA`` really is ``True`` or ``False``, since ``True & True`` is ``True``,
+but ``True & False`` is ``False``, so we can't determine the output.
+
+
+This differs from how ``np.nan`` behaves in logical operations. Pandas treated
+``np.nan`` is *always false in the output*.
+
+In ``or``
+
+.. ipython:: python
+
+   pd.Series([True, False, np.nan], dtype="object") | True
+   pd.Series([True, False, np.nan], dtype="boolean") | True
+
+In ``and``
+
+.. ipython:: python
+
+   pd.Series([True, False, np.nan], dtype="object") & True
+   pd.Series([True, False, np.nan], dtype="boolean") & True
+
+.. _Kleene Logic: https://en.wikipedia.org/wiki/Three-valued_logic#Kleene_and_Priest_logics
diff --git a/doc/source/user_guide/index.rst b/doc/source/user_guide/index.rst
index b86961a71433b..30b1c0b4eac0d 100644
--- a/doc/source/user_guide/index.rst
+++ b/doc/source/user_guide/index.rst
@@ -30,6 +30,7 @@ Further information on any specific method can be obtained in the
     missing_data
     categorical
     integer_na
+    boolean
     visualization
     computation
     groupby
diff --git a/pandas/core/arrays/boolean.py b/pandas/core/arrays/boolean.py
index aec3397bddd16..31dc656eb4b25 100644
--- a/pandas/core/arrays/boolean.py
+++ b/pandas/core/arrays/boolean.py
@@ -184,6 +184,9 @@ class BooleanArray(ExtensionArray, ExtensionOpsMixin):
     represented by 2 numpy arrays: a boolean array with the data and
     a boolean array with the mask (True indicating missing).
 
+    BooleanArray implements Kleene logic (sometimes called three-value
+    logic) for logical operations. See :ref:`boolean.kleene` for more.
+
     To construct an BooleanArray from generic array-like input, use
     :func:`pandas.array` specifying ``dtype="boolean"`` (see examples
     below).
@@ -283,7 +286,7 @@ def __getitem__(self, item):
 
     def _coerce_to_ndarray(self, dtype=None, na_value: "Scalar" = libmissing.NA):
         """
-        Coerce to an ndarary of object dtype or bool dtype (if force_bool=True).
+        Coerce to an ndarray of object dtype or bool dtype (if force_bool=True).
 
         Parameters
         ----------
@@ -565,10 +568,13 @@ def logical_method(self, other):
                 # Rely on pandas to unbox and dispatch to us.
                 return NotImplemented
 
+            assert op.__name__ in {"or_", "ror_", "and_", "rand_", "xor", "rxor"}
             other = lib.item_from_zerodim(other)
+            other_is_booleanarray = isinstance(other, BooleanArray)
+            other_is_scalar = lib.is_scalar(other)
             mask = None
 
-            if isinstance(other, BooleanArray):
+            if other_is_booleanarray:
                 other, mask = other._data, other._mask
             elif is_list_like(other):
                 other = np.asarray(other, dtype="bool")
@@ -576,22 +582,26 @@ def logical_method(self, other):
                     raise NotImplementedError(
                         "can only perform ops with 1-d structures"
                     )
-                if len(self) != len(other):
-                    raise ValueError("Lengths must match to compare")
                 other, mask = coerce_to_array(other, copy=False)
+            elif isinstance(other, np.bool_):
+                other = other.item()
+
+            if other_is_scalar and not (other is libmissing.NA or lib.is_bool(other)):
+                raise TypeError(
+                    "'other' should be pandas.NA or a bool. Got {} instead.".format(
+                        type(other).__name__
+                    )
+                )
 
-            # numpy will show a DeprecationWarning on invalid elementwise
-            # comparisons, this will raise in the future
-            with warnings.catch_warnings():
-                warnings.filterwarnings("ignore", "elementwise", FutureWarning)
-                with np.errstate(all="ignore"):
-                    result = op(self._data, other)
+            if not other_is_scalar and len(self) != len(other):
+                raise ValueError("Lengths must match to compare")
 
-            # nans propagate
-            if mask is None:
-                mask = self._mask
-            else:
-                mask = self._mask | mask
+            if op.__name__ in {"or_", "ror_"}:
+                result, mask = ops.kleene_or(self._data, other, self._mask, mask)
+            elif op.__name__ in {"and_", "rand_"}:
+                result, mask = ops.kleene_and(self._data, other, self._mask, mask)
+            elif op.__name__ in {"xor", "rxor"}:
+                result, mask = ops.kleene_xor(self._data, other, self._mask, mask)
 
             return BooleanArray(result, mask)
 
diff --git a/pandas/core/ops/__init__.py b/pandas/core/ops/__init__.py
index d14fb040c4e30..f3c01efed6d43 100644
--- a/pandas/core/ops/__init__.py
+++ b/pandas/core/ops/__init__.py
@@ -39,6 +39,7 @@
     _op_descriptions,
 )
 from pandas.core.ops.invalid import invalid_comparison  # noqa:F401
+from pandas.core.ops.mask_ops import kleene_and, kleene_or, kleene_xor  # noqa: F401
 from pandas.core.ops.methods import (  # noqa:F401
     add_flex_arithmetic_methods,
     add_special_arithmetic_methods,
diff --git a/pandas/core/ops/dispatch.py b/pandas/core/ops/dispatch.py
index c39f4d6d9698d..016a89eb56da3 100644
--- a/pandas/core/ops/dispatch.py
+++ b/pandas/core/ops/dispatch.py
@@ -189,6 +189,9 @@ def maybe_dispatch_ufunc_to_dunder_op(
         "ge",
         "remainder",
         "matmul",
+        "or",
+        "xor",
+        "and",
     }
     aliases = {
         "subtract": "sub",
@@ -204,6 +207,9 @@ def maybe_dispatch_ufunc_to_dunder_op(
         "less_equal": "le",
         "greater": "gt",
         "greater_equal": "ge",
+        "bitwise_or": "or",
+        "bitwise_and": "and",
+        "bitwise_xor": "xor",
     }
 
     # For op(., Array) -> Array.__r{op}__
diff --git a/pandas/core/ops/mask_ops.py b/pandas/core/ops/mask_ops.py
new file mode 100644
index 0000000000000..fd91e78451da9
--- /dev/null
+++ b/pandas/core/ops/mask_ops.py
@@ -0,0 +1,178 @@
+"""
+Ops for masked ararys.
+"""
+from typing import Optional, Union
+
+import numpy as np
+
+from pandas._libs import lib, missing as libmissing
+
+
+def kleene_or(
+    left: Union[bool, np.ndarray],
+    right: Union[bool, np.ndarray],
+    left_mask: Optional[np.ndarray],
+    right_mask: Optional[np.ndarray],
+):
+    """
+    Boolean ``or`` using Kleene logic.
+
+    Values are NA where we have ``NA | NA`` or ``NA | False``.
+    ``NA | True`` is considered True.
+
+    Parameters
+    ----------
+    left, right : ndarray, NA, or bool
+        The values of the array.
+    left_mask, right_mask : ndarray, optional
+        The masks. Only one of these may be None, which implies that
+        the associated `left` or `right` value is a scalar.
+
+    Returns
+    -------
+    result, mask: ndarray[bool]
+        The result of the logical or, and the new mask.
+    """
+    # To reduce the number of cases, we ensure that `left` & `left_mask`
+    # always come from an array, not a scalar. This is safe, since because
+    # A | B == B | A
+    if left_mask is None:
+        return kleene_or(right, left, right_mask, left_mask)
+
+    assert isinstance(left, np.ndarray)
+
+    raise_for_nan(right, method="or")
+
+    if right is libmissing.NA:
+        result = left.copy()
+    else:
+        result = left | right
+
+    if right_mask is not None:
+        # output is unknown where (False & NA), (NA & False), (NA & NA)
+        left_false = ~(left | left_mask)
+        right_false = ~(right | right_mask)
+        mask = (
+            (left_false & right_mask)
+            | (right_false & left_mask)
+            | (left_mask & right_mask)
+        )
+    else:
+        if right is True:
+            mask = np.zeros_like(left_mask)
+        elif right is libmissing.NA:
+            mask = (~left & ~left_mask) | left_mask
+        else:
+            # False
+            mask = left_mask.copy()
+
+    return result, mask
+
+
+def kleene_xor(
+    left: Union[bool, np.ndarray],
+    right: Union[bool, np.ndarray],
+    left_mask: Optional[np.ndarray],
+    right_mask: Optional[np.ndarray],
+):
+    """
+    Boolean ``xor`` using Kleene logic.
+
+    This is the same as ``or``, with the following adjustments
+
+    * True, True -> False
+    * True, NA   -> NA
+
+    Parameters
+    ----------
+    left, right : ndarray, NA, or bool
+        The values of the array.
+    left_mask, right_mask : ndarray, optional
+        The masks. Only one of these may be None, which implies that
+        the associated `left` or `right` value is a scalar.
+
+    Returns
+    -------
+    result, mask: ndarray[bool]
+        The result of the logical xor, and the new mask.
+    """
+    if left_mask is None:
+        return kleene_xor(right, left, right_mask, left_mask)
+
+    raise_for_nan(right, method="xor")
+    if right is libmissing.NA:
+        result = np.zeros_like(left)
+    else:
+        result = left ^ right
+
+    if right_mask is None:
+        if right is libmissing.NA:
+            mask = np.ones_like(left_mask)
+        else:
+            mask = left_mask.copy()
+    else:
+        mask = left_mask | right_mask
+
+    return result, mask
+
+
+def kleene_and(
+    left: Union[bool, libmissing.NAType, np.ndarray],
+    right: Union[bool, libmissing.NAType, np.ndarray],
+    left_mask: Optional[np.ndarray],
+    right_mask: Optional[np.ndarray],
+):
+    """
+    Boolean ``and`` using Kleene logic.
+
+    Values are ``NA`` for ``NA & NA`` or ``True & NA``.
+
+    Parameters
+    ----------
+    left, right : ndarray, NA, or bool
+        The values of the array.
+    left_mask, right_mask : ndarray, optional
+        The masks. Only one of these may be None, which implies that
+        the associated `left` or `right` value is a scalar.
+
+    Returns
+    -------
+    result, mask: ndarray[bool]
+        The result of the logical xor, and the new mask.
+    """
+    # To reduce the number of cases, we ensure that `left` & `left_mask`
+    # always come from an array, not a scalar. This is safe, since because
+    # A | B == B | A
+    if left_mask is None:
+        return kleene_and(right, left, right_mask, left_mask)
+
+    assert isinstance(left, np.ndarray)
+    raise_for_nan(right, method="and")
+
+    if right is libmissing.NA:
+        result = np.zeros_like(left)
+    else:
+        result = left & right
+
+    if right_mask is None:
+        # Scalar `right`
+        if right is libmissing.NA:
+            mask = (left & ~left_mask) | left_mask
+
+        else:
+            mask = left_mask.copy()
+            if right is False:
+                # unmask everything
+                mask[:] = False
+    else:
+        # unmask where either left or right is False
+        left_false = ~(left | left_mask)
+        right_false = ~(right | right_mask)
+        mask = (left_mask & ~right_false) | (right_mask & ~left_false)
+
+    return result, mask
+
+
+def raise_for_nan(value, method):
+    if lib.is_float(value) and np.isnan(value):
+        raise ValueError(f"Cannot perform logical '{method}' with floating NaN")
diff --git a/pandas/tests/arrays/test_boolean.py b/pandas/tests/arrays/test_boolean.py
index a13bb8edc8e48..d2f1a4a60ada8 100644
--- a/pandas/tests/arrays/test_boolean.py
+++ b/pandas/tests/arrays/test_boolean.py
@@ -356,6 +356,13 @@ def test_ufunc_reduce_raises(values):
 
 
 class TestLogicalOps(BaseOpsUtil):
+    def test_numpy_scalars_ok(self, all_logical_operators):
+        a = pd.array([True, False, None], dtype="boolean")
+        op = getattr(a, all_logical_operators)
+
+        tm.assert_extension_array_equal(op(True), op(np.bool(True)))
+        tm.assert_extension_array_equal(op(False), op(np.bool(False)))
+
     def get_op_from_name(self, op_name):
         short_opname = op_name.strip("_")
         short_opname = short_opname if "xor" in short_opname else short_opname + "_"
@@ -368,43 +375,205 @@ def get_op_from_name(self, op_name):
 
         return op
 
-    def _compare_other(self, data, op_name, other):
-        op = self.get_op_from_name(op_name)
-
-        # array
-        result = pd.Series(op(data, other))
-        expected = pd.Series(op(data._data, other), dtype="boolean")
+    def test_empty_ok(self, all_logical_operators):
+        a = pd.array([], dtype="boolean")
+        op_name = all_logical_operators
+        result = getattr(a, op_name)(True)
+        tm.assert_extension_array_equal(a, result)
 
-        # fill the nan locations
-        expected[data._mask] = np.nan
+        result = getattr(a, op_name)(False)
+        tm.assert_extension_array_equal(a, result)
 
-        tm.assert_series_equal(result, expected)
+        # TODO: pd.NA
+        # result = getattr(a, op_name)(pd.NA)
+        # tm.assert_extension_array_equal(a, result)
 
-        # series
-        s = pd.Series(data)
-        result = op(s, other)
+    def test_logical_length_mismatch_raises(self, all_logical_operators):
+        op_name = all_logical_operators
+        a = pd.array([True, False, None], dtype="boolean")
+        msg = "Lengths must match to compare"
 
-        expected = pd.Series(data._data)
-        expected = op(expected, other)
-        expected = pd.Series(expected, dtype="boolean")
+        with pytest.raises(ValueError, match=msg):
+            getattr(a, op_name)([True, False])
 
-        # fill the nan locations
-        expected[data._mask] = np.nan
+        with pytest.raises(ValueError, match=msg):
+            getattr(a, op_name)(np.array([True, False]))
 
-        tm.assert_series_equal(result, expected)
+        with pytest.raises(ValueError, match=msg):
+            getattr(a, op_name)(pd.array([True, False], dtype="boolean"))
 
-    def test_scalar(self, data, all_logical_operators):
+    def test_logical_nan_raises(self, all_logical_operators):
         op_name = all_logical_operators
-        self._compare_other(data, op_name, True)
+        a = pd.array([True, False, None], dtype="boolean")
+        msg = "Got float instead"
 
-    def test_array(self, data, all_logical_operators):
-        op_name = all_logical_operators
-        other = pd.array([True] * len(data), dtype="boolean")
-        self._compare_other(data, op_name, other)
-        other = np.array([True] * len(data))
-        self._compare_other(data, op_name, other)
-        other = pd.Series([True] * len(data), dtype="boolean")
-        self._compare_other(data, op_name, other)
+        with pytest.raises(TypeError, match=msg):
+            getattr(a, op_name)(np.nan)
+
+    @pytest.mark.parametrize("other", ["a", 1])
+    def test_non_bool_or_na_other_raises(self, other, all_logical_operators):
+        a = pd.array([True, False], dtype="boolean")
+        with pytest.raises(TypeError, match=str(type(other).__name__)):
+            getattr(a, all_logical_operators)(other)
+
+    def test_kleene_or(self):
+        # A clear test of behavior.
+        a = pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        b = pd.array([True, False, None] * 3, dtype="boolean")
+        result = a | b
+        expected = pd.array(
+            [True, True, True, True, False, None, True, None, None], dtype="boolean"
+        )
+        tm.assert_extension_array_equal(result, expected)
+
+        result = b | a
+        tm.assert_extension_array_equal(result, expected)
+
+        # ensure we haven't mutated anything inplace
+        tm.assert_extension_array_equal(
+            a, pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        )
+        tm.assert_extension_array_equal(
+            b, pd.array([True, False, None] * 3, dtype="boolean")
+        )
+
+    @pytest.mark.parametrize(
+        "other, expected",
+        [
+            (pd.NA, [True, None, None]),
+            (True, [True, True, True]),
+            (np.bool_(True), [True, True, True]),
+            (False, [True, False, None]),
+            (np.bool_(False), [True, False, None]),
+        ],
+    )
+    def test_kleene_or_scalar(self, other, expected):
+        # TODO: test True & False
+        a = pd.array([True, False, None], dtype="boolean")
+        result = a | other
+        expected = pd.array(expected, dtype="boolean")
+        tm.assert_extension_array_equal(result, expected)
+
+        result = other | a
+        tm.assert_extension_array_equal(result, expected)
+
+        # ensure we haven't mutated anything inplace
+        tm.assert_extension_array_equal(
+            a, pd.array([True, False, None], dtype="boolean")
+        )
+
+    def test_kleene_and(self):
+        # A clear test of behavior.
+        a = pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        b = pd.array([True, False, None] * 3, dtype="boolean")
+        result = a & b
+        expected = pd.array(
+            [True, False, None, False, False, False, None, False, None], dtype="boolean"
+        )
+        tm.assert_extension_array_equal(result, expected)
+
+        result = b & a
+        tm.assert_extension_array_equal(result, expected)
+
+        # ensure we haven't mutated anything inplace
+        tm.assert_extension_array_equal(
+            a, pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        )
+        tm.assert_extension_array_equal(
+            b, pd.array([True, False, None] * 3, dtype="boolean")
+        )
+
+    @pytest.mark.parametrize(
+        "other, expected",
+        [
+            (pd.NA, [None, False, None]),
+            (True, [True, False, None]),
+            (False, [False, False, False]),
+            (np.bool_(True), [True, False, None]),
+            (np.bool_(False), [False, False, False]),
+        ],
+    )
+    def test_kleene_and_scalar(self, other, expected):
+        a = pd.array([True, False, None], dtype="boolean")
+        result = a & other
+        expected = pd.array(expected, dtype="boolean")
+        tm.assert_extension_array_equal(result, expected)
+
+        result = other & a
+        tm.assert_extension_array_equal(result, expected)
+
+        # ensure we haven't mutated anything inplace
+        tm.assert_extension_array_equal(
+            a, pd.array([True, False, None], dtype="boolean")
+        )
+
+    def test_kleene_xor(self):
+        a = pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        b = pd.array([True, False, None] * 3, dtype="boolean")
+        result = a ^ b
+        expected = pd.array(
+            [False, True, None, True, False, None, None, None, None], dtype="boolean"
+        )
+        tm.assert_extension_array_equal(result, expected)
+
+        result = b ^ a
+        tm.assert_extension_array_equal(result, expected)
+
+        # ensure we haven't mutated anything inplace
+        tm.assert_extension_array_equal(
+            a, pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        )
+        tm.assert_extension_array_equal(
+            b, pd.array([True, False, None] * 3, dtype="boolean")
+        )
+
+    @pytest.mark.parametrize(
+        "other, expected",
+        [
+            (pd.NA, [None, None, None]),
+            (True, [False, True, None]),
+            (np.bool_(True), [False, True, None]),
+            (np.bool_(False), [True, False, None]),
+        ],
+    )
+    def test_kleene_xor_scalar(self, other, expected):
+        a = pd.array([True, False, None], dtype="boolean")
+        result = a ^ other
+        expected = pd.array(expected, dtype="boolean")
+        tm.assert_extension_array_equal(result, expected)
+
+        result = other ^ a
+        tm.assert_extension_array_equal(result, expected)
+
+        # ensure we haven't mutated anything inplace
+        tm.assert_extension_array_equal(
+            a, pd.array([True, False, None], dtype="boolean")
+        )
+
+    @pytest.mark.parametrize(
+        "other", [True, False, pd.NA, [True, False, None] * 3],
+    )
+    def test_no_masked_assumptions(self, other, all_logical_operators):
+        # The logical operations should not assume that masked values are False!
+        a = pd.arrays.BooleanArray(
+            np.array([True, True, True, False, False, False, True, False, True]),
+            np.array([False] * 6 + [True, True, True]),
+        )
+        b = pd.array([True] * 3 + [False] * 3 + [None] * 3, dtype="boolean")
+        if isinstance(other, list):
+            other = pd.array(other, dtype="boolean")
+
+        result = getattr(a, all_logical_operators)(other)
+        expected = getattr(b, all_logical_operators)(other)
+        tm.assert_extension_array_equal(result, expected)
+
+        if isinstance(other, BooleanArray):
+            other._data[other._mask] = True
+            a._data[a._mask] = False
+
+            result = getattr(a, all_logical_operators)(other)
+            expected = getattr(b, all_logical_operators)(other)
+            tm.assert_extension_array_equal(result, expected)
 
 
 class TestComparisonOps(BaseOpsUtil):