I will describe some classical and some new unobstructedeness theorems for the deformations of ordinary or non-commutative Calabi-Yau varieties. In particular I will explain a recent result on the smoothness of the moduli space of categories matrix factorizations. The result is based on some new Hodge theory and leads to an interesting Hodge theoretic invvariants of Landau-Ginzburg models. This is joint work with Katzarkov and Kontsevich.