  Elements of a global operator approach to the WZWN theory for compact Riemann
surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine
Krichever-Novikov algebras over a dense open subset of the moduli space of
Riemann surfaces (respectively of smooth, projective complex curves) with $N$
marked points are introduced. It is shown that the tangent space of the moduli
space at an arbitrary moduli point is isomorphic to a certain subspace of the
Krichever-Novikov vector field algebra given by the data of the moduli point.
This subspace is complementary to the direct sum of the two subspaces
containing the vector fields which vanish at the marked points, respectively
which are regular at a fixed reference point. For each representation of the
affine algebra $3g-3+N$ equations $(\partial_k+T[e_k])\Phi=0$ are given, where
the elements $\{e_k\}$ are a basis of the subspace, and $T$ is the affine
Sugawara representation of the centrally extended vector field algebra. For
genus zero one obtains the Knizhnik-Zamolodchikov equations in this way. The
coefficients of the equations for genus one are found in terms of
Weierstra\ss-$\sigma$ function.
