Proofs of Tsygans formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygans formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevichs formality quasi-isomorphism for Hochschild cochains of R[[y_1, >..., y_d]] and Shoikhets formality quasi-isomorphism for Hochschild chains of R[[y_1,..., y_d]] I prove Tsygans formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, abla), where M is the manifold and abla is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.
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