A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. Currently, we know only a very small percentage of the total cohomology. In this talk I will explain some of what is known and discuss some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups.