Environmental problems such as floods require statistical analysis that takes into account the complex nature of the data, namely observations are sampled at different spatial points in a given region for a certain time. Thus the spatial dependence structure cannot be ignored. Extreme statistics for the design of structures for flood protection, for the study of the structural failures such as bridges, dams, etc., for the prediction of heat waves and others should be based on a solid theoretical framework. Max-stable processes provide a such theory and in the last decade have emerged as fertile ground for research and a common tool for the statistical modeling of spatial extremes. This entry provides a summary of max-stable processes.