The structure of Hamiltonian symmetry reductions of the Wess-Zumino-Novikov-Witten (WZNW) theories by first class Kac-Moody (KM) constraints is analysed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability, conformal invariance and W-symmetry in the reduced theories. A Lagrangean, gauged WZNW implementation of the reduction is established in the general case and thereby the path integral as well as the BRST formalism are set up for studying the quantum version of the reduction. The general results are applied to a number of examples. In particular, a W-algebra is associated to each embedding of si B) into the simple Lie algebras by using purely first class constraints. The primary fields of these W-algebras are manifestly given by the siB) embeddings, but it is also shown that there is an siB) embedding present in every polynomial and primary KM reduction and that the PFn'-algebras have a hidden slB) structure too. New generalized Toda theories are found whose chiral algebras are the W-algebras based on the half-integral siB) embeddings, and the W-symmetry of the effective action of those generalized Toda theories associated with the integral gradings is exhibited explicitly.