We study deformation quantization of Poisson algebraic varieties. Using the universal deformation formulas of Kontsevich, and an algebro-geometric approach to the bundle of formal coordinate systems over a smooth variety X, we prove existence of deformation quantization of the sheaf of functions O_X (assuming the vanishing of certain cohomologies). Under slightly stronger assumptions we can classify all such deformations. Eprint: math.AG/0310399