For self-adjoint differential operators, a classical result of H.Weyl and its extensions since a century, give a very general law for the asymptotic distribution of eigenvalues in the high energy and in the semi-classical limits. This law describes the density of eigenvalues in terms of the direct image of the symplectic volume element on (the real) phase space under a symbol map. On the other hand, for differential operators in one dimension with analytic coefficients, the complex WKB-method is a powerful tool to study the spectrum. In the non-self-adjoint case, this often leads to a complex Bohr-Sommerfeld rule for the eigenvalues that invalidates the Weyl law. In higher dimensions, similar results can be obtained when the coefficients are analytic and the complex WKB-method is then replaced by analytic microlocal analysis and deformations of the real phase space. In 2 dimensions, one often encounters a “hidden” complete integrability, related to an old theorem of Cherry. Small non-self-adjoint perturbations of self-adjoint differential operators have their spectra in thin bands and quite detailed results are available in the integrable case. Non-self-adjoint operators do most of the time exhibit large resolvent norms and corresponding eigenvalue instability. It is then natural to study the effect of small random perturbations. It came as a surprise that such perturbations cause the eigenvalues to distribute according to the Weyl law. An heuristic explanation is that the random perturbations destroy analyticity and forbid complex deformations of phase space, leaving the Weyl law as the only natural possibility. We will state some results and develop the underlying ideas of complex phase space deformations and random perturbations, following works of A. Melin, M. Hitrik, S. Vu Ngoc, M. Hager, W. Bordeaux Montrieux, the speaker and other people.