The collection of representations of a given finite group has a rich structure as a tensor category. A fusion category is a tensor category which "looks like" the category of representations of a finite group. In addition to finite groups, fusion categories appear in the study of operator algebras, quantum groups, and conformal field theory. Quadratic fusion categories are a class of fusion categories which can be thought of as (categorical) quadratic extensions of finite groups. They play a prominent role in the classification of small-index subfactors in the theory of von Neumann algebras. In this talk we will explain what quadratic fusion categories are, describe some examples, and discuss some open questions about existence and classification of families of quadratic fusion categories.