  In this work, we study the asymptotic geometry of the mapping class group and
Teichmueller space. We introduce tools for analyzing the geometry of
`projection' maps from these spaces to curve complexes of subsurfaces; from
this we obtain information concerning the topology of their asymptotic cones.
We deduce several applications of this analysis. One of which is that the
asymptotic cone of the mapping class group of any surface is tree-graded in the
sense of Drutu and Sapir; this tree-grading has several consequences including
answering a question of Drutu and Sapir concerning relatively hyperbolic
groups. Another application is a generalization of the result of Brock and Farb
that for low complexity surfaces Teichmueller space, with the Weil-Petersson
metric, is delta-hyperbolic. Although for higher complexity surfaces these
spaces are not delta-hyperbolic, we establish the presence of previously
unknown negative curvature phenomena in the mapping class group and
Teichmueller space for arbitrary surfaces.
