Abstract: Measurable Group Theory is a recent term, referring to the study of properties of infinite discrete groups, preserved under a certain equivalence relation (called by Gromov Measure Equivalence). It is parallel in some ways to Geometric Group Theory, is closely related to Ergodic Theory, and has applications to such areas as von Neumann algebras, and Descriptive Set Theory (Logic).

In the talk I will give an overview of the field and will discuss some of the many recent exciting results concerning rigidity and geometric invariants. Groups to be featured include: lattices in Lie groups (e.g. SL(n,Z)), Mapping Class Groups, Kazhdan groups; invariants: amenability, property (T), L^2-Betti numbers, simplicial volume etc.