A theorem of Erdős and Rado generalizes Ramsey's theorem to infinite cardinals: for each cardinal n, there exists a cardinal N so that each graph with N vertices contains either a clique or an independent set of size n. In the infinite case, one can take n = N if n is countable but in most other uncountable cases N must be much bigger than n. Stability theory is a branch of model theory studying certain definability conditions allowing us to take n = N for a large number of infinite cardinals. Historically, stability theory was first developed by Shelah for classes axiomatized by a first-order theory. In this talk, I describe a generalization to a large class of categories, accessible categories. I will also talk about recent progress on the eventual categoricity conjecture, resolved by Morley and Shelah for first-order but still open for accessible categories. Speaker(s): Sebastien Vasey (Harvard University)