Star-products (one parameter formal deformations of the usual product on functions) serve as local models for DQ-algebroids. A DQ-algebroid is a formal one-parameter deformation its "classical limit" which in general is a twisted form of the structure sheaf of the the manifold.
As is well known, a star-product on functions on a manifold gives rise to a Poisson on the sheaf of functions. I will explain what sort of additional structure arises on the classical limit of a DQ-algebroid generalizing and extending the Poisson structure.