The aim of the paper is to derive spectral estimates into several
classes of magnetic systems. They include three-dimensional regions with
Dirichlet boundary as well as a particle in R-3 confined by a local
change of the magnetic field. We establish two-dimensional
Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic
fields and, using them, get three-dimensional estimates for the
eigenvalue moments of the corresponding magnetic Laplacians.