A conventional view of the spatial spread of invasive species dating back to the works by Fisher (1937) and Kolmogorov et al. (1937) is that it occurs via the propagation of a travelling population front. In a realistic 2D system, such a front normally separates the invaded area behind the front from the uninvaded areas in front of the front. This view has eventually been challenged by discovering an alternative scenario called “patchy invasion� where the spread takes place via the spatial dynamics of separate patches of high population density with a very low density between them, and a continuous population front does not exist at any time. Patchy invasion was studied theoretically in much detail using diffusion-reaction models. However, diffusion-reaction models have many limitations; in particular, they almost completely ignore long-distance dispersal. In this talk, I will present some new results showing that patchy invasion can occur as well when long-distance dispersal is taken into account. Mathematically, the system is described by integral-difference equations with fat-tailed dispersal kernels. I will also show that apparently minor details of kernel parametrization may have a relatively strong effect on the rate of species spread, which evokes the general issues of understanding the uncertainty and the limits of predictability in ecology.