correlation functions of primary fields using Ward identities, as was the case for the Virasoro algebra. A W(3)-Verma module appears in the fusion product of two other W(3)-Verma modules with a multiplicity that is in general infinite. Differential equations A correlation function may obey a differential equation that generalizes the BPZ equations if the fields have sufficiently many vanishing null vectors. A four-point function of W(N)-primary fields on the sphere with one fully degenerate field obeys a differential equation if but not if . In the latter case, for a differential equation to exist, one of the other fields must have vanishing null vectors. For example, a four-point function with two fields of momentums (fully degenerate) and with (almost fully degenerate) obeys a differential equation whose solutions are generalized hypergeometric functions of type . Applications to conformal field theory W-minimal models W-minimal models are generalizations of Virasoro minimal models based on a W-algebra. Their spaces of states are made of finitely many fully degenerate representations. They exist for certain rational values of the central charge: in the case of the W(N) algebra, values of the type A W(N)-minimal model with central charge may be constructed as a coset of Wess-Zumino-Witten models . For example, the two-dimensional critical three-state Potts model has central charge . Spin observables of the model may be described in terms of the D-series non-diagonal Virasoro minimal model with , or in terms of the diagonal W(3)-minimal model with . Conformal Toda theory Conformal Toda theory is a generalization of Liouville theory that is based on a W-algebra. Given a simple Lie algebra , the Lagrangian is a functional of a field which belongs to the root space of , with one interaction term for each simple root: This depends on the cosmological constant , which plays no meaningful role, and on the parameter , which is related to the central charge. The resulting field theory is a conformal field theory, whose chiral symmetry algebra is a W-algebra constructed from by Drinfeld-Sokolov reduction. For the preservation of conformal symmetry in the quantum theory, it is crucial that there are no more interaction terms than components of the vector . The methods that lead to the solution of Liouville theory may be applied to W(N)-conformal Toda theory, but they only lead to the analytic determination of a particular class of three-point structure constants, and W(N)-conformal Toda theory with