When studying geometric problems, it is often useful to produce some algebraic invariant that controls the geometry what we are interested in. Sometimes, these algebraic tools are of independent interest, and become topic of study. Hochschild (co)homology is an (Morita) invariant of an algebra A which carries a lot of structure and information. In the talk, we will define Hochschild (co)homology of a bimodule M over a k-algebra A, we will compute it in low degrees, and see what happens when we assume our algebra to be smooth over a field of characteristic zero. After doing so, we’ll deal with some explicit examples, and we’ll see that Hochschild homology (sometimes) carries a new differential. This observation will be the motivation to introduce the so called “Hodge-to-de Rham” spectral sequence, and state an open conjecture due to Kontsevich and Soibelman.