This is a survey on the proof of a generalized version of the Mumford conjecture obtained in joint work with M. Weiss stating that a certain map between some classifying spaces which a priori have different natures induces an isomorphism at the level of integral homology. We also discuss our proof of the original Mumford conjecture stating that the stable rational cohomology of the moduli space of Riemann surfaces is a certain polynomial algebra generated by the Mumford–Morita–Miller cohomology classes of even degrees.