Deligne's theorem in positive characteristic

Deligne proved that in characteristic zero, every tensor category of subexponential growth is super Tannakian. In this talk, I will survey recent work of Etingof, Ostrik et al aimed at extending this theorem to positive characteristic. I will introduce the Verlinde category, and explain why this is a counterexample to Deligne's theorem in char p >= 5. I don't plan to prove anything except basic lemmas and tautologies, but hope to provide the overarching narrative of this line of research with the help of several (counter-)examples and precise questions/results.