PDF Version The standard Cartan calculus on polyvector fields and exterior forms can be naturally extended to the Hochschild cohomology and the Hochschild homology of an arbitrary associative algebra $A$. Recent results of M. Kontsevich and Y. Soibelman imply that this calculus on Hochschild (co)homology can be naturally upgraded to a homotopy calculus structure on the pair $(C^{\bullet}(A), C_{\bullet}(A))$ ``Hochschild cochains $+$ Hochschild chains'' of an associative algebra $A$. In my talk I will consider the sheaf of homotopy calculi $(C^{\bullet}(\cO_X), C_{\bullet}(\cO_X))$ for a smooth algebraic variety $X$ with $\cO_X$ being the structure sheaf. I will show that this sheaf $(C^{\bullet}(\cO_X), C_{\bullet}(\cO_X))$ of homotopy calculi is quasi-isomorphic to its cohomology. I will also talk about applications of this result. My talk will be based on joint paper arXiv:0807.5117 with D. Tamarkin and B. Tsygan. The main result of this paper was announced by D. Tamarkin in 2000 at the Mosh\'e Flato memorial conference.