  We prove a homological stability theorem for moduli spaces of
simply-connected manifolds of dimension $2n > 4$, with respect to forming
connected sum with $S^n \times S^n$. This is analogous to Harer's stability
theorem for the homology of mapping class groups. Combined with previous work
of the authors, it gives a calculation of the homology of the moduli spaces of
manifolds diffeomorphic to connected sums of $S^n \times S^n$ in a range of
degrees.
