I will talk about definably amenable groups in NIP theories, that is NIP groups admitting a translation-invariant measure on the class of definable sets. Examples include stable groups, compact Lie groups and solvable NIP groups. In a joint work with Artem Chernikov, we characterize "generic types" in this context, describe the space of invariant measures and prove "generic compact domination". I will present this work and mention some more recent developments.

The aim of this tutorial is to present the model-theoretic tools used in applications to other areas of mathematics. I will focus on notions which will be employed in the other tutorials such as pseudofinite structures, stability, NIP, measures.