  It is shown by Connes, Douglas and Schwarz that gauge theory on
noncommutative torus describes compactifications of M-theory to tori with
constant background three-form field. This indicates that noncommutative gauge
theories on more general manifolds also can be useful in string theory. We
discuss a framework to noncommutative quantum gauge theory on Poisson manifolds
by using the deformation quantization. The Kontsevich formula for the star
product was given originally in terms of the perturbation expansion and it
leads to a non-renormalizable quantum field theory. We discuss the
nonperturbative path integral formulation of Cattaneo and Felder as a possible
approach to construction of noncommutative quantum gauge theory on Poisson
manifolds. Some other aspects of classical and quantum noncommutative field
theory are also discussed.
