We consider eigenvalues of matrices whose entries are Lipschitz continuous functions of a real parameter. In the case of Hermitian matrices, if the eigenvalues of such a matrix H(t) are plotted against t, it can typically be observed that the trajectories of the eigenvalue functions appear to collide; however, they undergo a last-minute repulsive effect, thus avoiding intersections .This phenomenon was first explained by Von Neumann and Wigner in 1929. I plan to describe the Neumann-Wigner approach, and then to expose our recent generalization to several different classes of matrices (beyond Hermitian).