This is the title of a talk given at the 2016 Logic Colloquium (University of Leeds) on August 4, 2016. The talk presented the corresponding two papers. 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. I will present my proof of the conjecture for

The proof combines Shelah's earlier work on universal classes with a study of AECs that have amalgamation and are *tame* (a locality property isolated by Grossberg and VanDieren which says roughly that orbital types are determined by their small restrictions).

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.