The mapping class group of a compact surface S is the group of homeomorphisms of S modulo isotopy. Via its Cayley graph it can be viewed as an infinite-diameter metric space, whose large-scale geometry is strongly connected with the algebraic properties of the group. In joint work with Behrstock, Kleiner and Mosher, we study this large-scale geometry and prove in particular that its quasi-isometries are bounded perturbations of the action of the group. This implies (as was also independently shown by Hamenstadt) that the group is quasi-isometrically rigid.