A most notable characteristic of non-Hermitian matrices is that their spectra can be intrinsically sensitive to tiny perturbation. Although this spectral instability causes the numerical analysis of their spectra to be extremely unreliable, it has recently been shown to be also the source of new mathematical phenomena. I will present recent results about the eigenvalues asymptotics and eigenvector localization for deterministic non-Hermitian Toeplitz matrices with small additive random perturbations. These results are related to recent developments in the theory of partial differential equations. The talk is based on joint work with J. Sjöstrand, and with A. Basak and O. Zeitouni.