In this talk we will review some recent results on random Schrödinger operators, which are used to model electronic transport in disordered quantum systems and to study the phenomenon of Anderson localization. After a short introduction to the subject, we will focus on a particular type of random operator driven by a fractional laplacian. The interest on the latter lies in their association to stable Levy processes, random walks with long jumps and anomalous diffusion. We will discuss in this talk the interplay between the non-locality of the fractional laplacian and the localization properties of the random potential in the fractional Anderson model, in both the continuous and discrete settings. In the discrete setting we study the integrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the probability community. This is based on joint work with M. Gebert (LMU Munich).