Shelah's categoricity conjecture: If an L_{\omega_1,\omega} theory is categorical in a cardinal greater than the Hanf number then the theory is categorical in every cardinal above the Hanf number. Despite many papers by Shelah and others, the conjecture is still open.

In the late seventies Shelah introduced the notion of abstract elementary class (a semantic generalization of L_{\omega_1,\omega} theory) and formulated a similar strong categoricity conjecture.

Recently Monica VanDieren and the speaker proved that Shelah's conjecture holds for a large family of abstract elementary classes. Our proof turned to be less technical than expected, one of the surprises is that it is in ZFC, while previous related results of Shelah make heavy use of diamond-like principles. Our argument is new even when one specialize to first-order logic.

In the talk I will describe all the notions in this abstract and the general framework. I intend to make my talk to be accessible also to people who did not take a course in model theory.