I will give an introduction to the infinite loop spaces associated to the mapping class groups of surfaces and to the automorphism groups of free groups and explain how automorphisms of free groups with boundaries can be used to show that the map between these two infinite loop spaces is an infinite loop map. These results make use of homological stability for mapping class groups (Harer-Ivanonv) and for automorphisms of free groups with boundaries (Hatcher-W.).