In this article we are interested in the numerical computation of spectra ofnon-self adjoint quadratic operators. This leads to solve nonlinear eigenvalue prob-lems. We begin with a review of theoretical results for the spectra of quadratic oper-ators, especially for the Schr ?dinger pencils. Then we present the numerical methods developed to compute the spectra: spectral methods and finite difference discretiza-tion, in infinite or in bounded domains. The numerical results obtained are analyzed and compared with the theoretical results. The main difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are very unstable.