Title: Homology of mapping class groups and Torelli groups

Abstract: Given a compact, oriented surface, its mapping class group is the group of isotopy classes of orientation preserving homeomorphisms of the surface. These groups are of great interest in topology, but also in algebraic, hyperbolic and complex geometry. A particularly interesting subgroup of the mapping class group is its Torelli group, consisting of those mapping classes that act trivially on the homology of the surface. In this talk, I will give a gentle introduction to mapping class groups, Torelli groups and the homology of these groups, finishing with some of my own work in this area, which mainly concerns Torelli groups of other low dimensional spaces than orientable surfaces.