I will discuss recent joint work with Oscar Randal-Williams, aimed at calculating the cohomology of BDiff(W) and related spaces, where W is a smooth 2n-dimensional manifold, Diff(W) is the topological group of diffeomorphisms of W, and BDiff(W) is its classifying space. Surprisingly, the cohomology ring turns out to be partially independent of W through a range of degrees (homological stability). In this talk, I will discuss how infinite loop spaces can be used to describe the cohomology in this stable range.