### Abstract

Consider a periodic Schroedinger operator with magnetic wells on a noncompact, simply connected, Riemannian manifold equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries. We will discuss sufficient conditions on the magnetic field, which ensure the existence of a gap (or, even more, an arbitrarily large number of gaps) in the spectrum of such an operator in the semi-classical limit. These results are partially based on a joint work with B. Helffer.