Abstract: We consider the spectral problem for the 2D Schroedinger operator for a charged particle in strong magnetic and periodic electric fields. The related classical problem is analyzed first by means of the Krylov-Bogoljubov-Alfven and Neishtadt averaging methods. It allows us to show ``almost integrability'' of the original 2D classical Hamilton system, and to reduce it to 1D one on the phase space which is the 2D torus. Topological methods for integrable Hamiltonian system and also elementary facts from the Morse theory give the general classification of the classical motion and some topological characteristics (like rotation numbers and Maslov indices). Using this classsification, and the semiclassical approximation we give the general asymptotic description of the (band) spectrum of the original Schroedinger operator and, in particular, estimates for the number of the bands in each Landau level.