We study reaction-diffusion equations $u_t = L u + f(u)$ with homogeneous reactions f and diffusion operators L arising from the theory of Levy processes, with emphasis on propagation phenomena. The classical diffusion case (L = Laplacian) has been well-studied, including questions about traveling fronts, wavefront propagation, existence of spreading speeds, etc. After a brief review of the one-dimensional theory, we will concentrate on the case of nonlocal diffusions in several dimensions. We will discuss questions concerning long time dynamics of solutions, including spreading vs. quenching and existence of spreading speeds.