Description It is well known that, for self-adjoint operators, asymptotic properties of spectra are deeply connected with real Lagrangian geometry and theory of Hamiltonian systems. For example, semi-classical eigenvalues can be computed from Maslov quantization conditions on Lagrangian manifolds; these manifolds have to be invariant with respect to Hamiltonian systems, defined by symbols of initial operators. For non-selfadjoint operators, the corresponding theory is not well developed. We describe known results in this direction (including quite recent ones); the main idea of the new theory is to replace the real geometry by the complex one and to describe spectral characteristics of operators via geometrical properties of complex manifolds. We study this correspondence for a number of operators, which are popular in mathematical physics, and discuss physical applications.