  This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a
general and flexible context for differential calculus in derived geometry,
including the de Rham algebra and polyvector fields. We then introduce the
formalism of formal derived stacks and prove formal localization and gluing
results. These allow us to define shifted Poisson structures on general derived
Artin stacks, and prove that the non-degenerate Poisson structures correspond
exactly to shifted symplectic forms. Shifted deformation quantization for a
derived Artin stack endowed with a shifted Poisson structure is discussed in
the last section. This paves the way for shifted deformation quantization of
many interesting derived moduli spaces, like those studied in [PTVV] and
probably many others.
