The theory and applications of max-stable processes have been developing rapidly over the last decades. Max-stable processes are important models in spatial extreme value theory since they arise as the only possible non-degenerate limits of maxima of iid processes under affine normalization. They are used to assess environmental risks in the framework of the annual maxima method. The purpose of these lectures is to provide an introduction to the theory of max-stable processes with a particular focus on a series of recent results obtained thanks to a careful analysis of de Haan's poisson point process representation of max-stable processes.