  The use of reaction-diffusion models rests on the key assumption that the
underlying diffusive process is Gaussian. However, a growing number of studies
have pointed out the prevalence of anomalous diffusion, and there is a need to
understand the dynamics of reactive systems in the presence of this type of
non-Gaussian diffusion. Here we present a study of front dynamics in
reaction-diffusion systems where anomalous diffusion is due to the presence of
asymmetric Levy flights. Our approach consists of replacing the Laplacian
diffusion operator by a fractional diffusion operator, whose fundamental
solutions are Levy $\alpha$-stable distributions. Numerical simulation of the
fractional Fisher-Kolmogorov equation, and analytical arguments show that
anomalous diffusion leads to the exponential acceleration of fronts and a
universal power law decay, $x^{-\alpha}$, of the tail, where $\alpha$, the
index of the Levy distribution, is the order of the fractional derivative.
