Proofs of Tsygan's formality conjectures for chains formulated in math.QA/9904132 would unlock important algebraic tools which might lead to new generalizations of the famous Atiyah-Singer index theorem and Riemann-Roch-Hirzebruch theorem. In the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my recent paper math.QA/0402248 I have proven Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R[[y_1, ... , y_d]] proposed in paper math.QA/0010321 by Shoikhet. In my talk I will formulate Tsygan's formality conjecture, outline the idea of the proof and say a couple of words about applications of this result to computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.