Let H be a Hopf algebra, and A, B be H-Galois extensions. We investigate the category AM H B of relative Hopf bimodules, and the Morita equivalences between A and B induced by them. Introduction This paper is a contribution to the representation theory of Hopf-Galois extensions, as originated by Schneider in [15]. More specifically, we consider the following questions. Let H be a Hopf algebra, ...