  We prove that the homology of the mapping class groups of non-orientable
surfaces stabilizes with the genus of the surface. Combining our result with
recent work of Madsen and Weiss, we obtain that the classifying space of the
stable mapping class group of non-orientable surfaces, up to homology
isomorphism, is the infinite loop space of a Thom spectrum build from the
canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular,
we show that the stable rational cohomology is a polynomial algebra on
generators in degrees 4i--this is the non-oriented analogue of the Mumford
conjecture.
