Appears in collection : Algèbre, Géométrie et Physique : une conférence en l'honneur

In this talk I will present the recent developments concerning the interactions between deformation quantization and derived algebraic geometry. In a first part I'll briefly recall some elements of derived algebraic geometry, including the notions of shifted symplectic and Poisson structures. In the second part, I will focus on the existence of deformation quantization of shifted symplectic and Poisson structures. I will mainly present two approaches, a first one based on a generalization of Kontsevich's formality, and a second one inspired by Fedosov's quantization of symplectic manifolds.