Monoidal categories include, for example, representations of finite groups and Lie algebras, categories of bimodules for a ring, and braided and symmetric tensor categories. There are powerful homological methods for understanding the structure of various types of monoidal categories, their objects, and deformations. In this introductory talk we will primarily focus on one such setting, that of Hochschild cohomology, while mentioning others in which similar methods work. Fairly generally, the cohomology of an (exact) monoidal category (that is, Ext of two copies of the unit object) has both a graded commutative multiplication and a graded Lie bracket, both important operations that facilitate understanding of deformation theory, give rise to support varieties for objects and tensor ideals, and more.