We consider a class of monostable nonlinear equations with nonlocal diffusion and local or nonlocal reaction in R^d. If the nonlocal diffusion is given through an anisotropic probability kernel decaying at least exponentially fast in a direction and if the initial condition has the similar property, then the solution propagates in this direction at most linearly in time. For a particular equation arising in population ecology, we study then traveling waves and describe the front of propagation. In contrast, if either the kernel or the initial condition (for the general equation) have appropriately regular heavy tails, then an acceleration of the front propagation for solutions is observed and studied. For the one-dimensional case we show also that the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may yield different orders of the acceleration.