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.