  In this paper, we formulate a generalization of the classical BRST
construction which applies to the case of the reduction of a poisson manifold
by a submanifold. In the case of symplectic reduction, our procedure
generalizes the usual classical BRST construction which only applies to
symplectic reduction of a symplectic manifold by a coisotropic submanifold,
\ie\ the case of reducible ``first class'' constraints. In particular, our
procedure yields a method to deal with ``second-class'' constraints. We
construct the BRST complex and compute its cohomology. BRST cohomology vanishes
for negative dimension and is isomorphic as a poisson algebra to the algebra of
smooth functions on the reduced poisson manifold in zero dimension. We then
show that in the general case of reduction of poisson manifolds, BRST
cohomology cannot be identified with the cohomology of vertical differential
forms.
