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These are Lecture Notes of a course given by the author at the School Winter Braids, held at the Université de Pau et des Pays de L’Adour (France), on February 2015. It is explained how mapping class groups, and in particular braid groups, act on some interesting geometric spaces like the hyperbolic plane and the complex of curves, and how this allows to obtain some algebraic properties of the groups. A proof of the hyperbolicity of the graph of curves, following Hensel-Przytycki-Webb, is given.

Geometric approaches to braid groups and mapping class groups