The mapping class group is the group of topological symmetries of a surface.
By understanding the homology and cohomology of the mapping class group and its subgroups,
we gain insight into its finiteness properties (finite generation, finite presentability, etc.)
and we can also classify topological invariants of surface bundles. In this talk,
we will introduce basic notions about the mapping class group and explain how to compute its
low dimensional homology groups. Then, we will explain some recent work with Mladen Bestvina and
Kai-Uwe Bux concerning the homology of the Torelli subgroup of the mapping class group,
the group of elements acting trivially on the homology of the surface. In particular, we answer
a question of Mess by proving that the cohomological dimension of the Torelli group for
a genus g surface is 3g-5. At the end, we will present some computer calculations designed to
help us understand the more complicated homological properties of the Torelli groups.