  In this paper, we study the spreading properties of the solutions of an
integro-differential equation of the form $u_t=J\ast u-u+f(u).$ We focus on
equations with slowly decaying dispersal kernels $J(x)$ which correspond to
models of population dynamics with long-distance dispersal events. We prove
that for kernels $J$ which decrease to $0$ slower than any exponentially
decaying function, the level sets of the solution $u$ propagate with an
infinite asymptotic speed. Moreover, we obtain lower and upper bounds for the
position of any level set of $u.$ These bounds allow us to estimate how the
solution accelerates, depending on the kernel $J$: the slower the kernel
decays, the faster the level sets propagate. Our results are in sharp contrast
with most results on this type of equation, where the dispersal kernels are
generally assumed to decrease exponentially fast, leading to finite propagation
speeds.
