I will discuss new invariants and problems for definable groups in first order theories given by notions from  topological dynamics. This is work with and/or by   Chernikov, Gismatullin, Krupinski, Penazzi, Simon,

The Canonical Base Property (CBP for short) was introduced by Pillay, inspired by results in Complex geometry. It implies the dichotomy "modular/fields" for rank 1 types, and is satisfied by the two main examples of fields with operators. After giving the definition, I will discuss how one can prove it, some of its consequences, notably a descent result. I will also mention some applications to algebraic questions.