  We consider a magnetic Schr\"odinger operator $H^h$, depending on a
semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that
there is no electric field. We suppose that the minimal value $b_0$ of the
intensity of the magnetic field $b$ is strictly positive. We give a survey of
the results on asymptotic behavior of the eigenvalues of the operator $H^h$ in
the semiclassical limit.
