  We determine the moduli space of classical solutions to the field equations
of Poisson Sigma Models on arbitrary Riemann surfaces for Poisson structures
with vanishing Poisson form class. This condition ensures the existence of a
presymplectic form on the target Poisson manifold which agrees with the induced
symplectic forms of the Poisson tensor upon pullback to the leaves. The
dimension of the classical moduli space as a function of the genus of the
worldsheet Sigma and the corank k of the Poisson tensor is determined as
k(rank(H^1(Sigma))+1). Representatives of the classical solutions are provided
using the above mentioned presymplectic 2-forms, and possible generalizations
to cases where such a form does not exist are discussed. The results are
compared to the known moduli space of classical solutions for two-dimensional
BF and Yang-Mills theories.
