The process of desertification can be modeled by systems of reaction-diffusion equations. Numerical simulations of these models agree remarkably well with field observations: both show that 'vegetation patterns'—i.e. regions in which the vegetation only survives in localized 'patches'—naturally appear as the transition between a healthy homogeneously vegetated state and the (non-vegetated) desert state. Desertification is a catastrophic and non-reversible event during which huge patterned vegetation areas 'collapse' into the desert state at a fast time scale—for instance as a consequence of a slow decrease of yearly rainfall, or through an increased grazing pressure. It is crucial to be able to recognize whether a patterned state is close to collapse (or not), ecologists are thus searching for 'early warning signals.' In this talk, we will translate the issues raised by the desertification process into mathematical terms and relate these to recent developments in the field of pattern formation. It will be shown that the process of desertification poses fundamental mathematical questions and that it already has led to the development of novel mathematical theory.