  We propose a new approach to the study of the correlation functions of
W-algebras. The conformal blocks (chiral correlation functions), for fixed
arguments, are defined to be those linear functionals on the product of the
highest weight (h.w.) representation spaces which satisfy the Ward identities.
First we investigate the dimension of the chiral correlation functions in the
case when there is no singular vector in any of the representations. Then we
pass to the analysis of the completely degenerate representations. A special
subspace of the h.w. representation spaces, introduced by Nahm, plays an
important role in the considerations. The structure of these subspaces shows a
deep connection with the quantum and classical Toda models and relates certain
completely degenerate representations of the \wg algebra to representations of
$G$. This is confirmed by an analysis for the Virasoro, \wa2 and \wbc2 algebra.
We also relate our work to Nahms, Feigen-Fuchs' and Watts' results.
