  We study theories with W-algebra symmetries and their relation to WZNW models
on (super-)groups. Correlation functions of the WZNW models are expressed in
terms of correlators of CFTs with W-algebra symmetry. The symmetries of the
theories involved in these correspondences are related by the Drinfeld-Sokolov
reduction of Lie algebras to W-algebras. The W-algebras considered in this
paper are the Bershadsky-Polyakov algebra for sl(3) and the
quasi-superconformal algebra for generic sl(N|M). The quantum W-algebras
obtained from affine sl(N) are constructed using embeddings of sl(2) into
sl(N), and these can in turn be characterized by partitions of N. The above
cases correspond to \underline{N+2} = \underline{2} + N \underline{1} and its
supergroup extension. Finally, sl(2N) and the correspondence corresponding to
\underline{2N} = N \underline{2} is also analyzed.
