Non-self-adjoint operators appear in many settings, from kinetic theory 
and quantum mechanics to linearizations of equations of mathematical 
physics. The spectral analysis of such operators, while often notoriously 
difficult, reveals a wealth of new phenomena, compared with their 
self-adjoint counterparts. Spectra for non-self-adjoint operators display 
fascinating features, such as lattices of eigenvalues for operators of 
Kramers-Fokker-Planck type, say, and eigenvalues for operators with 
analytic coefficients in dimension one, concentrated to unions of curves 
in the complex spectral plane. In this talk, after a general introduction, 
we shall discuss spectra for non-self-adjoint perturbations of 
self-adjoint operators in dimension two, under the assumption that the 
classical flow of the unperturbed part is completely integrable.
The role played by the flow-invariant Lagrangian tori of the completely 
integrable system, both Diophantine and rational, in the spectral analysis 
of the non-self-adjoint operators will be described. In particular, we 
shall discuss the spectral contributions of rational tori, leading to 
eigenvalues having the form of the "legs in a spectral centipede". This 
talk is based on joint work with Johannes Sj\"ostrand.