This is the title of a talk given at the Seminář z algebry at Masaryk university on March 16, 2017. The talk presents the corresponding two papers.

Universal classes are a general model-theoretic framework introduced in the seventies by Saharon Shelah to study certain classes of modules. It encompasses several natural algebraic objects, such as vector spaces and locally finite groups.

I will present generalizations to universal classes of several concepts and results of linear algebra. For example, a universal class which has a single model of a "high-enough" infinite size has a single model in every high-enough size. Moreover, such classes admit an independence notion generalizing linear independence in vector spaces. I will also discuss a more general framework (also due to Shelah), abstract elementary classes, and conjectured extensions of these results there.

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, vol. 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*, Accepted, Annals of Pure and Applied Logic. Preprint: pdf arXiv.Sebastien Vasey,

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

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

*A survey on tame abstract elementary classes*, Accepted, Beyond First Order Model Theory (José Iovino ed.), CRC Press. Preprint: pdf arXiv.