It is sometimes useful to systematically study Abelian categories equipped with some additional structure, e.g., tensor categories. I will talk about some less well known families of examples, Abelian categorifications, which appear in the representation theory of classical groups and related algebras. The simplest situation is that of a Heisenberg categorification, which is an Abelian category equipped with extra structure making it into a module category over a (monoidal) Heisenberg category. Heisenberg categorifications are ubiquitous in many parts of type A representation theory, and a remarkably rich structure theory has emerged which involves also the parallel notion of a Kac-Moody categorification. This has its origins in various works of Chuang, Khovanov, Lauda and Rouquier. It is the most highly developed GL branch of a more general story. There are also emerging OSp, P and Q branches associated to the other three classical families of supergroups over algebraically closed fields.