We shall present recent results giving precise eigenvalue asymptotics for the magnetic Laplacian in the semiclassical limit, in the case of a confining, non-uniform field, in dimensions 2 and 3. The main ingredients are the symplectic geometry of the magnetic characteristic manifold and the quantization of the various oscillations of the Hamiltonian flow for long times. This is joint work with B. Helffer, Y. Kordyukov and San Vu Ngoc.