I’ll discuss recent work on the structure of Heisenberg categories and their generalizations, especially the higher-level Heisenberg categorifications defined by Brundan.  This includes the long sought-after identification of their Grothendieck groups with Heisenberg algebras, and a construction of categorical Kac-Moody actions from Heisenberg actions which unifies essentially all examples of categorical Kac-Moody actions of interest to representation theorists.  This is joint work with Jonathan Brundan and Alistair Savage.