This will be a mostly-expository talk on topological approaches to the unstable cohomology of the moduli space M_g of smooth genus g curves. The solution by Madsen-Weiss of the Mumford conjecture means we understand the stable cohomology (the limit as g -> infinity) completely -- it consists wholly of tautological classes -- and Faber's conjectures give a conjectural picture of the tautological cohomology for finite g as well. However, the unstable cohomology (all the rest!) remains completely unknown.
I'll talk about some constructions from topology that, although quite non-algebraic, are central to understanding the non-tautological cohomology of M_g. This includes a version of Poincare duality for M_g, where the curve complex plays the role that modular symbols play for SL_n(Z), as well as certain "abelian cycles" determined by commuting Dehn twists. I'll finish with some new vanishing theorems for the top cohomology of M_g and a conjectural picture of a large chunk of the remaining unstable cohomology (both joint with Benson Farb and Andrew Putman)