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This talk will be meant as an introduction to the growing area of statistical mechanics on graphs and more specifically to percolation on graphs. The emphasis will be on graphs that can be seen as models of crystals in a non-Euclidean space.

Among the most interesting features of statistical mechanics on "non-Euclidean graphs" is the splitting of critical points, giving rise to regimes and phenomena that are absent in the "Euclidean case". In addition, the critical behavior is expected then to be of mean-field nature, and this conjecture has been proved to hold in extreme cases.