Spectral properties of non-self-adjoint operators

This text is a slightly expanded version of my 6 hour mini-course at the PDE-meeting in 'Evian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations.