  We describe a canonical reduction of AKSZ-BV theories to the cohomology of
the source manifold. We get a finite dimensional BV theory that describes the
contribution of the zero modes to the full QFT. Integration can be defined and
correlators can be computed. As an illustration of the general construction we
consider two dimensional Poisson sigma model and three dimensional Courant
sigma model. When the source manifold is compact, the reduced theory is a
generalization of the AKSZ construction where we take as source the cohomology
ring. We present the possible generalizations of the AKSZ theory.
