Derived categories of Calabi-Yau manifolds have been very actively studied recently due to M. Kontsevich's suggestion that they form the framework for describing the physicists categories of branes (boundary conditions for open strings). Physics has, however, a more general construction of such categories: the Landau-Ginzburg models, which Kontsevich argued is related to categories of matrix factorizations, studied by Buchweitz and Eisenbud in the '80's. In my talk I shall explain how matrix factorizations can be regarded as twisted complexes over certain A_infty algebras, a fact which allows us to compute the Hochschild homology of their category under certain reasonable assumptions. This is joint work with my student Junwu Tu.