  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 $\mathbb{R}^3$ 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.
