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This paper continues a geometric study of Harvey's Complex of Curves, whoseultimate goal is to apply the theory of hyperbolic spaces and groups toalgorithmic questions for the Mapping Class Group and geometric properties ofKleinian representations. The authors' previous result that the complex isdelta-hyperbolic was hard to apply because the complex is not locally finite;in this paper some tools are developed for overcoming this problem, and acombinatorial mechanism introduced which describes sequences of elementarymoves in the graph of markings on a surface. These tools are applied to give afamily of quasi-geodesic words in the Mapping Class Group, and a linear boundon the shortest word conjugating two conjugate pseudo-Anosov elements.  A basic tool in the analysis is a family of subsurface projections, which areroughly analogous to closest-point projections to horoballs in classicalhyperbolic space. These projections have a strong contraction property whichmakes it possible to tie together the geometry of the complex and that of the(infinite) subcomplexes that arise as links of vertices. The resulting layeredstructure of the complex is controlled by means of a combinatorial devicecalled a hierarchy of geodesics, which is the central construction of thepaper.

Geometry of the complex of curves II: Hierarchical structure