The interaction between infinitary logic and the model theory of abstract elementary classes has had a serious imprint of large cardinals since the inception of AECs. Although later developments in AECs have emphasized a more purely model theoretic treatment, capturing independence-like relations, there are various fundamental questions on the relation between various logics and AECs — and, in some of these, large cardinals are central.

I will discuss some work by Boney on these connections, as well as some recent joint work by Väänänen and myself.