A basic question about any group or space is what its (rational) cohomology groups look like. I will give a summary of what is known about such cohomology of the mapping class group of an oriented surface, and then discuss the vast amount of cohomology Putman and I discovered for some of its finite index subgroups. This has consequences for the coherent cohomological dimension of the moduli space of curves, which I will outline. (Joint work with Andy Putman)