In characteristic zero, the structure theory of (symmetric) tensor categories (with objects of finite length) is governed by Deligne’s classical results: a tensor category is the representation category of a supergroup if and only if it is of moderate growth. On the other hand, in characteristic p>0 there was very little progress until the last few years. One of the major developments is the notion of incompressible categories, which take over the role played by the category of (super)vector spaces in characteristic zero. In this talk, I’ll give a broad overview of the classical results, more recent work (mostly not by me), and open conjectures. In other words: what’s known, what’s unknown, and why do non-category theorists care?