Following a work by A. Melin and J. Sj\"ostrand, it has become
increasingly clear that non-selfadjoint operators in dimension two share many of the pleasant features of operators in dimension one. In particular, in the semiclassical limit, it is often possible to get complete asymptotics for individual eigenvalues of such operators in some domain in the complex plane, by means of a suitable Bohr-Sommerfeld quantization rule. In this talk, we would like to report on some recent results in this direction obtained together with Johannes Sj\"ostrand, as
a part of an ongoing program on small non-selfadjoint perturbations of selfadjoint operators. We shall also try to discuss applications to asymptotics of scattering poles for semiclassical Schr\"odinger operators, and to dissipative wave equations on compact manifolds.