The homotopy type of the cobordism category

We define cobordism categories of manifolds, and show that their classifying spaces are weakly homotopy equivalent to infinite loop spaces arising from Thom spectra. We generalise this to manifolds with tangential structures, for example oriented manifolds or manifolds with maps to a fixed space X. As an application, a new and shorter proof is given of the generalised Mumford conjecture concerning mapping class groups of surfaces.