diff --git a/pymc/distributions/__init__.py b/pymc/distributions/__init__.py
index bb91ea23c2a..2aff0c18c03 100644
--- a/pymc/distributions/__init__.py
+++ b/pymc/distributions/__init__.py
@@ -21,6 +21,7 @@
 )
 
 from pymc.distributions.bound import Bound
+from pymc.distributions.censored import Censored
 from pymc.distributions.continuous import (
     AsymmetricLaplace,
     Beta,
@@ -187,6 +188,7 @@
     "Rice",
     "Moyal",
     "Simulator",
+    "Censored",
     "CAR",
     "PolyaGamma",
     "logpt",
diff --git a/pymc/distributions/censored.py b/pymc/distributions/censored.py
new file mode 100644
index 00000000000..14b92f30359
--- /dev/null
+++ b/pymc/distributions/censored.py
@@ -0,0 +1,146 @@
+#   Copyright 2020 The PyMC Developers
+#
+#   Licensed under the Apache License, Version 2.0 (the "License");
+#   you may not use this file except in compliance with the License.
+#   You may obtain a copy of the License at
+#
+#       http://www.apache.org/licenses/LICENSE-2.0
+#
+#   Unless required by applicable law or agreed to in writing, software
+#   distributed under the License is distributed on an "AS IS" BASIS,
+#   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#   See the License for the specific language governing permissions and
+#   limitations under the License.
+import aesara.tensor as at
+import numpy as np
+
+from aesara.scalar import Clip
+from aesara.tensor import TensorVariable
+from aesara.tensor.random.op import RandomVariable
+
+from pymc.distributions.distribution import SymbolicDistribution, _get_moment
+from pymc.util import check_dist_not_registered
+
+
+class Censored(SymbolicDistribution):
+    r"""
+    Censored distribution
+
+    The pdf of a censored distribution is
+
+    .. math::
+
+        \begin{cases}
+            0 & \text{for } x < lower, \\
+            \text{CDF}(lower, dist) & \text{for } x = lower, \\
+            \text{PDF}(x, dist) & \text{for } lower < x < upper, \\
+            1-\text{CDF}(upper, dist) & \text {for} x = upper, \\
+            0 & \text{for } x > upper,
+        \end{cases}
+
+
+    Parameters
+    ----------
+    dist: PyMC unnamed distribution
+        PyMC distribution created via the .dit() API, which will be censored. This
+        distribution must be univariate and have a logcdf method implemeted.
+    lower: float or None
+        Lower (left) censoring point. If `None` the distribution will not be left censored
+    upper: float or None
+        Upper (right) censoring point. If `None`, the distribution will not be right censored.
+
+
+    Examples
+    --------
+    .. code-block:: python
+
+        with pm.Model():
+            normal_dist = pm.Normal.dist(mu=0.0, sigma=1.0)
+            censored_normal = pm.Censored("censored_normal", normal_dist, lower=-1, upper=1)
+    """
+
+    @classmethod
+    def dist(cls, dist, lower, upper, **kwargs):
+        if not isinstance(dist, TensorVariable) or not isinstance(dist.owner.op, RandomVariable):
+            raise ValueError(
+                f"Censoring dist must be a distribution created via the `.dist()` API, got {type(dist)}"
+            )
+        if dist.owner.op.ndim_supp > 0:
+            raise NotImplementedError(
+                "Censoring of multivariate distributions has not been implemented yet"
+            )
+        check_dist_not_registered(dist)
+        return super().dist([dist, lower, upper], **kwargs)
+
+    @classmethod
+    def rv_op(cls, dist, lower=None, upper=None, size=None, rngs=None):
+        if lower is None:
+            lower = at.constant(-np.inf)
+        if upper is None:
+            upper = at.constant(np.inf)
+
+        # Censoring is achieved by clipping the base distribution between lower and upper
+        rv_out = at.clip(dist, lower, upper)
+
+        # Reference nodes to facilitate identification in other classmethods, without
+        # worring about possible dimshuffles
+        rv_out.tag.dist = dist
+        rv_out.tag.lower = lower
+        rv_out.tag.upper = upper
+
+        if size is not None:
+            rv_out = cls.change_size(rv_out, size)
+        if rngs is not None:
+            rv_out = cls.change_rngs(rv_out, rngs)
+
+        return rv_out
+
+    @classmethod
+    def ndim_supp(cls, *dist_params):
+        return 0
+
+    @classmethod
+    def change_size(cls, rv, new_size):
+        dist_node = rv.tag.dist.owner
+        lower = rv.tag.lower
+        upper = rv.tag.upper
+        rng, old_size, dtype, *dist_params = dist_node.inputs
+        new_dist = dist_node.op.make_node(rng, new_size, dtype, *dist_params).default_output()
+        return cls.rv_op(new_dist, lower, upper)
+
+    @classmethod
+    def change_rngs(cls, rv, new_rngs):
+        (new_rng,) = new_rngs
+        dist_node = rv.tag.dist.owner
+        lower = rv.tag.lower
+        upper = rv.tag.upper
+        olg_rng, size, dtype, *dist_params = dist_node.inputs
+        new_dist = dist_node.op.make_node(new_rng, size, dtype, *dist_params).default_output()
+        return cls.rv_op(new_dist, lower, upper)
+
+    @classmethod
+    def graph_rvs(cls, rv):
+        return (rv.tag.dist,)
+
+
+@_get_moment.register(Clip)
+def get_moment_censored(op, rv, dist, lower, upper):
+    moment = at.switch(
+        at.eq(lower, -np.inf),
+        at.switch(
+            at.isinf(upper),
+            # lower = -inf, upper = inf
+            0,
+            # lower = -inf, upper = x
+            upper - 1,
+        ),
+        at.switch(
+            at.eq(upper, np.inf),
+            # lower = x, upper = inf
+            lower + 1,
+            # lower = x, upper = x
+            (lower + upper) / 2,
+        ),
+    )
+    moment = at.full_like(dist, moment)
+    return moment
diff --git a/pymc/distributions/distribution.py b/pymc/distributions/distribution.py
index cf9d87eb0b3..86a1c1e7fbd 100644
--- a/pymc/distributions/distribution.py
+++ b/pymc/distributions/distribution.py
@@ -26,6 +26,7 @@
 from aeppl.logprob import _logcdf, _logprob
 from aesara import tensor as at
 from aesara.tensor.basic import as_tensor_variable
+from aesara.tensor.elemwise import Elemwise
 from aesara.tensor.random.op import RandomVariable
 from aesara.tensor.random.var import RandomStateSharedVariable
 from aesara.tensor.var import TensorVariable
@@ -628,6 +629,12 @@ def get_moment(rv: TensorVariable) -> TensorVariable:
     return _get_moment(rv.owner.op, rv, *rv.owner.inputs).astype(rv.dtype)
 
 
+@_get_moment.register(Elemwise)
+def _get_moment_elemwise(op, rv, *dist_params):
+    """For Elemwise Ops, dispatch on respective scalar_op"""
+    return _get_moment(op.scalar_op, rv, *dist_params)
+
+
 class Discrete(Distribution):
     """Base class for discrete distributions"""
 
diff --git a/pymc/tests/test_distributions.py b/pymc/tests/test_distributions.py
index 0095638e576..efe42006f45 100644
--- a/pymc/tests/test_distributions.py
+++ b/pymc/tests/test_distributions.py
@@ -3275,3 +3275,73 @@ def logp(value, mu):
         ).shape
         == to_tuple(size)
     )
+
+
+class TestCensored:
+    @pytest.mark.parametrize("censored", (False, True))
+    def test_censored_workflow(self, censored):
+        # Based on pymc-examples/censored_data
+        rng = np.random.default_rng(1234)
+        size = 500
+        true_mu = 13.0
+        true_sigma = 5.0
+
+        # Set censoring limits
+        low = 3.0
+        high = 16.0
+
+        # Draw censored samples
+        data = rng.normal(true_mu, true_sigma, size)
+        data[data <= low] = low
+        data[data >= high] = high
+
+        with pm.Model(rng_seeder=17092021) as m:
+            mu = pm.Normal(
+                "mu",
+                mu=((high - low) / 2) + low,
+                sigma=(high - low) / 2.0,
+                initval="moment",
+            )
+            sigma = pm.HalfNormal("sigma", sigma=(high - low) / 2.0, initval="moment")
+            observed = pm.Censored(
+                "observed",
+                pm.Normal.dist(mu=mu, sigma=sigma),
+                lower=low if censored else None,
+                upper=high if censored else None,
+                observed=data,
+            )
+
+            prior_pred = pm.sample_prior_predictive()
+            posterior = pm.sample(tune=500, draws=500)
+            posterior_pred = pm.sample_posterior_predictive(posterior)
+
+        expected = True if censored else False
+        assert (9 < prior_pred.prior_predictive.mean() < 10) == expected
+        assert (13 < posterior.posterior["mu"].mean() < 14) == expected
+        assert (4.5 < posterior.posterior["sigma"].mean() < 5.5) == expected
+        assert (12 < posterior_pred.posterior_predictive.mean() < 13) == expected
+
+    def test_censored_invalid_dist(self):
+        with pm.Model():
+            invalid_dist = pm.Normal
+            with pytest.raises(
+                ValueError,
+                match=r"Censoring dist must be a distribution created via the",
+            ):
+                x = pm.Censored("x", invalid_dist, lower=None, upper=None)
+
+        with pm.Model():
+            mv_dist = pm.Dirichlet.dist(a=[1, 1, 1])
+            with pytest.raises(
+                NotImplementedError,
+                match="Censoring of multivariate distributions has not been implemented yet",
+            ):
+                x = pm.Censored("x", mv_dist, lower=None, upper=None)
+
+        with pm.Model():
+            registered_dist = pm.Normal("dist")
+            with pytest.raises(
+                ValueError,
+                match="The dist dist was already registered in the current model",
+            ):
+                x = pm.Censored("x", registered_dist, lower=None, upper=None)
diff --git a/pymc/util.py b/pymc/util.py
index 92f2462e1fb..f2183688540 100644
--- a/pymc/util.py
+++ b/pymc/util.py
@@ -332,3 +332,20 @@ def cf(self):
         return cf
 
     return cachedmethod(self_cache_fn(f.__name__), key=hash_key)(f)
+
+
+def check_dist_not_registered(dist, model=None):
+    """Check that a dist is not registered in the model already"""
+    from pymc.model import modelcontext
+
+    try:
+        model = modelcontext(None)
+    except TypeError:
+        pass
+    else:
+        if dist in model.basic_RVs:
+            raise ValueError(
+                f"The dist {dist} was already registered in the current model.\n"
+                f"You should use an unregistered (unnamed) distribution created via "
+                f"the `.dist()` API instead, such as:\n`dist=pm.Normal.dist(0, 1)`"
+            )