  We prove Strichartz estimates for the Schroedinger equation with an
electromagnetic potential, in dimension $n\geq3$. The decay and regularity
assumptions on the potentials are almost critical, i.e., close to the Coulomb
case. In addition, we require repulsivity and a non trapping condition, which
are expressed as smallness of suitable components of the potentials. However,
the potentials themselves can be large, and we avoid completely any a priori
spectral assumption on the operator. The proof is based on smoothing estimates
and new Sobolev embeddings for spaces associated to magnetic potentials.
