Biological invasion admittedly consists of a few distinctly different stages such as exotic species introduction, establishment and geographical spread. Each of the stages has its own specific mechanisms and implications, which require application of specific research approaches. In my talk, I focus on the challenges arising during the stage of the geographical spread. A well-developed theory based on diffusion-reaction equations predicts a simple pattern of alien species spread consisting of a continuous traveling boundary or 'population front' separating the invaded and non-invaded regions. A propagating population front has been a paradigm of the invasive species spread for several decades. However, it also appears to be at odds with some observations. In some cases, the spread takes place through formation of a distinct patchy spatial structure without any continuous boundary. Perhaps the most well known and well studied example of this 'patchy invasion' is the gypsy moth spread in the USA; e.g. see www.fs.fed.us/ne/morgantown/4557/gmoth/atlas/#defoliation In order to address this problem theoretically, I first re-examine the current views on possible mechanisms of the patchy spread and argue that the importance of the stratified diffusion may be significantly overestimated. Second, I will revisit the traditional diffusion-reaction framework and show that the patchy spread is, in fact, its inherent property in case the invasive species is affected by predation or an infectious disease and its growth is damped by the strong Allee effect. The patchy spread described by a diffusion-reaction model appears to be a scenario of alien species invasion "at the edge of extinction" and this can have important implications for the management and control of the invasive species. I will then show that patchy spread is not an exclusive property of the diffusion-reaction systems but can be observed as well in a completely different type of model such as a coupled map lattice which is capable of taking into account environmental heterogeneity. Finally, I will argue that these theoretical results taken together with the evidence from field data may result in a paradigm shift: A typical pattern of exotic species spread is a patchy invasion rather than the continuous population front propagation.