diff --git a/distreqx/distributions/__init__.py b/distreqx/distributions/__init__.py
index 06b8aa3..7cdb660 100644
--- a/distreqx/distributions/__init__.py
+++ b/distreqx/distributions/__init__.py
@@ -10,6 +10,7 @@
     AbstractSurvivalDistribution as AbstractSurvivalDistribution,
 )
 from ._gamma import Gamma as Gamma
+from ._improper_uniform import ImproperUniform as ImproperUniform
 from ._independent import Independent as Independent
 from ._logistic import Logistic as Logistic
 from ._mixture_same_family import MixtureSameFamily as MixtureSameFamily
diff --git a/distreqx/distributions/_improper_uniform.py b/distreqx/distributions/_improper_uniform.py
new file mode 100644
index 0000000..eb73470
--- /dev/null
+++ b/distreqx/distributions/_improper_uniform.py
@@ -0,0 +1,82 @@
+import jax.numpy as jnp
+from jaxtyping import Array, Key
+
+from ._distribution import AbstractDistribution
+
+
+class ImproperUniform(
+    AbstractDistribution,
+    strict=True,
+):
+    """Improper Uniform distribution over the entire real line.
+
+    This distribution has an unnormalized probability density of 1 everywhere,
+    meaning its `log_prob` evaluates to 0. As an improper distribution, it
+    does not integrate to 1 and cannot be sampled.
+    """
+
+    shape: tuple[int, ...] = ()
+
+    @property
+    def event_shape(self) -> tuple[int, ...]:
+        return self.shape
+
+    def sample(self, key: Key[Array, ""]) -> Array:
+        """Sampling is not defined for improper distributions."""
+        raise NotImplementedError("Cannot sample from an ImproperUniform distribution.")
+
+    def sample_and_log_prob(self, key: Key[Array, ""]) -> tuple[Array, Array]:
+        raise NotImplementedError("Cannot sample from an ImproperUniform distribution.")
+
+    def log_prob(self, value: Array) -> Array:
+        """Returns the unnormalized log probability (constant 0.0)."""
+        return jnp.zeros_like(value)
+
+    def prob(self, value: Array) -> Array:
+        """Returns the unnormalized probability (constant 1.0)."""
+        return jnp.ones_like(value)
+
+    def entropy(self) -> Array:
+        """Entropy of an improper uniform over the reals is infinite."""
+        return jnp.full(self.shape, jnp.inf)
+
+    def icdf(self, value: Array) -> Array:
+        raise NotImplementedError("icdf is undefined for an improper distribution.")
+
+    def log_cdf(self, value: Array) -> Array:
+        raise NotImplementedError("log_cdf is undefined for an improper distribution.")
+
+    def cdf(self, value: Array) -> Array:
+        raise NotImplementedError("cdf is undefined for an improper distribution.")
+
+    def survival_function(self, value: Array) -> Array:
+        raise NotImplementedError(
+            "survival_function is undefined for an improper distribution."
+        )
+
+    def log_survival_function(self, value: Array) -> Array:
+        raise NotImplementedError(
+            "log_survival_function is undefined for an improper distribution."
+        )
+
+    def mean(self) -> Array:
+        raise NotImplementedError("Mean is undefined for an improper distribution.")
+
+    def median(self) -> Array:
+        raise NotImplementedError("Median is undefined for an improper distribution.")
+
+    def variance(self) -> Array:
+        raise NotImplementedError("Variance is undefined for an improper distribution.")
+
+    def stddev(self) -> Array:
+        raise NotImplementedError(
+            "Standard deviation is undefined for an improper distribution."
+        )
+
+    def mode(self) -> Array:
+        raise NotImplementedError("Mode is undefined for an improper distribution.")
+
+    def kl_divergence(self, other_dist, **kwargs) -> Array:
+        raise NotImplementedError(
+            "KL divergence is undefined for an improper distribution."
+        )
diff --git a/docs/api/distributions/improper_uniform.md b/docs/api/distributions/improper_uniform.md
new file mode 100644
index 0000000..b096c2b
--- /dev/null
+++ b/docs/api/distributions/improper_uniform.md
@@ -0,0 +1,4 @@
+# Improper Uniform
+
+::: distreqx.distributions.ImproperUniform
+---
\ No newline at end of file
diff --git a/mkdocs.yml b/mkdocs.yml
index aa5cf4e..d63bc07 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -111,6 +111,7 @@ nav:
             - 'api/distributions/categorical.md'
             - 'api/distributions/gamma.md'
             - 'api/distributions/logistic.md'
+            - 'api/distributions/improper_uniform.md'
             - 'api/distributions/independent.md'
             - 'api/distributions/mixture_same_family.md'
             - 'api/distributions/uniform.md'