  In this paper we consider the structure of general quantum W-algebras. We
introduce the notions of deformability, positive-definiteness, and reductivity
of a W-algebra. We show that one can associate a reductive finite Lie algebra
to each reductive W-algebra. The finite Lie algebra is also endowed with a
preferred $sl(2)$ subalgebra, which gives the conformal weights of the
W-algebra. We extend this to cover W-algebras containing both bosonic and
fermionic fields, and illustrate our ideas with the Poisson bracket algebras of
generalised Drinfeld-Sokolov Hamiltonian systems. We then discuss the
possibilities of classifying deformable W-algebras which fall outside this
class in the context of automorphisms of Lie algebras. In conclusion we list
the cases in which the W-algebra has no weight one fields, and further, those
in which it has only one weight two field.
