  In this paper we prove that any Poisson structure on a sheaf of Lie
algebroids admits a weak deformation quantization, and give a sufficient
condition for such a Poisson structure to admit an actual deformation
quantization. We also answer the corresponding classification problems. In the
complex symplectic case, we recover in particular some results of Nest-Tsygan
and Polesello-Schapira.
  We begin the paper with a recollection of known facts about deformation
theory of cosimplicial differential graded Lie algebras.
