This is the title of a talk given at 2016 Workshop on set-theoretical aspects of the model theory of strong logics, at the Centre de Recerca Matemàtica (Universitat Autònoma de Barcelona) on September 29, 2016. Here are the slides.

Abstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an *L _{∞, ω}* theory, as well as numerous algebraic examples. They were introduced by Saharon Shelah forty years ago. Shelah focused on generalizations of Morley's categoricity theorem and conjectured the following eventual version: An AEC categorical in a high-enough cardinal is categorical on a tail of cardinals. Historically, most approximations to the conjecture have assumed categoricity in a

With the assumption of categoricity in a successor and assuming the amalgamation property, Shelah proved a downward transfer while Grossberg and VanDieren have proven an upward transfer assuming in addition a locality property that they called *tameness*. Recently, Boney has shown that tameness follows from a large cardinal axiom, obtaining (assuming large cardinals) Shelah's eventual categoricity conjecture when the categoricity cardinal is a successor.

In this talk, we will discuss the situation when the categoricity cardinal is limit. In this case an additional property, having prime models over sets of the form *Ma*, plays an important role. In particular (assuming large cardinals) Shelah's eventual categoricity conjecture holds in AECs that have prime models. Here, the large cardinals are only used to obtain amalgamation and tameness, and in some instances they can also be derived model-theoretically. For example, the categoricity conjecture holds (without assuming large cardinals) for classes of models of a *universal* *L _{∞, ω}* theory.

John T. Baldwin,

*Categoricity*, University Lecture Series, vol. 50, American Mathematical Society, 2009.Saharon Shelah,

*Classification Theory for Abstract Elementary Classes*, Studies in Logic: Mathematical Logic and foundations, vols. 18 and 20, College Publications, 2009.Sebastien Vasey,

*The lazy model theoretician's guide to Shelah's eventual categoricity conjecture in universal classes*, An expository note: pdf arXiv.Sebastien Vasey,

*Shelah's eventual categoricity conjecture in universal classes. Part I*, Preprint: pdf arXiv.Sebastien Vasey,

*Shelah's eventual categoricity conjecture in universal classes. Part II*, Preprint: pdf arXiv.Sebastien Vasey,

*Shelah's eventual categoricity conjecture in tame AECs with primes*, Preprint: pdf arXiv.Will Boney and Sebastien Vasey,

*A survey on tame abstract elementary classes*, Preprint: pdf arXiv.