  We study the large scale geometry of mapping class groups MCG(S), using
hyperbolicity properties of curve complexes. We show that any self
quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance
away from a left-multiplication, and as a consequence obtain quasi-isometric
rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually
equal to it. (The latter theorem was proved by Hamenstadt using different
methods). As part of our approach we obtain several other structural results: a
description of the tree-graded structure on the asymptotic cone of MCG(S); a
characterization of the image of the curve-complex projection map from MCG(S)
to the product of the curve complexes of essential subsurfaces of S; and a
construction of Sigma-hulls in MCG(S), an analogue of convex hulls.
