Abstract: (joint work with Allen Hatcher) Let M be a compact, connected 3-manifold with a fixed boundary component d_0M. For each prime manifold P, we consider the mapping class group of the manifold M_n^P obtained from M by taking a connected sum with n copies of P. We prove that the ith homology of this mapping class group is independent of n in the range n>2i+1. Our theorem moreover applies to certain subgroups of the mapping class group and include, as special cases, homological stability for the automorphism groups of free groups and of other free products, for the symmetric groups and for wreath products with symmetric groups.