I will present several results concerning the Lifschitz-tail asymptotics for operators of the form L=H+V, where H is a (possibly nonlocal) Levy operator and V is a random potential of either Poisson or lattice-type. We prove the existence of Lifschitz tail and establish the precise rate of decay of the integrated density of states near the bottom of the spectrum. The results were obtained in collaboration with Kamil Kaleta (Wroclaw University of Science and Technology).