Deformation quantization deals with non-commutative deformations of symplectic (or more generally Poisson) algebraic varieties. I this talk I will present a general approach to the subject based on derived algebraic geometry, and explain how it can be used in order to treat new examples of quantum objects of geometric origin (e.g. moduli spaces of sheaves on CY varieties). I will present the recent results and progress, as well as some open questions.