  Let $\mathcal{C}$ be a finite tensor category and $\mathcal{M}$ an exact left
$\mathcal{C}$-module category. We call $\mathcal{M}$ unimodular if the finite
multitensor category ${\sf Rex}_{\mathcal{C}}(\mathcal{M})$ of right exact
$\mathcal{C}$-module endofunctors of $\mathcal{M}$ is unimodular. In this
article, we provide various characterizations, properties, and examples of
unimodular module categories. As our first application, we employ unimodular
module categories to construct (commutative) Frobenius algebra objects in the
Drinfeld center of any finite tensor category. When $\mathcal{C}$ is a pivotal
category, and $\mathcal{M}$ is a unimodular, pivotal left $\mathcal{C}$-module
category, the Frobenius algebra objects are symmetric as well. Our second
application is a classification of unimodular module categories over the
category of finite dimensional representations of a finite dimensional Hopf
algebra; this answers a question of Shimizu. Using this, we provide an example
of a finite tensor category whose categorical Morita equivalence class does not
contain any unimodular tensor category.
