To a compact smooth manifold M is associated a so called moduli space, having several properties of interest. First it is a classifying space for smooth fiber bundles with fibers isomorphic to that given manifold, so that its cohomology gives rise to characteristic classes for manifold bundles. But also, its homotopy type is closely related to that of the diffeomorphism group of M, Diff(M), one of the main characters in manifold topology, and still not fully understood as of today even for the simplest case where the manifold is the disc D^n (if n>=4).
I will introduce the works of Galatius/Madsen/Tillmann/Weiss and Galatius/Randal-Williams which compute the cohomology of these moduli spaces in even dimensions, in a range of degrees."