  This text is a survey of recent results obtained by the author and
collaborators on different problems for non-self-adjoint operators. The topics
are: Kramers-Fokker-Planck type operators, spectral asymptotics in two
dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint
operators with small random perturbations. In the introduction we also review
the notion of pseudo-spectrum and its relation to non-self-adjoint spectral
problems.
