Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. Which manifolds have such a self-similar structure? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.