  We define noncommutative deformations $W_q^s(G)$ of algebras of functions on
certain (finite coverings of) transversal slices to the set of conjugacy
classes in an algebraic group $G$ which play the role of Slodowy slices in
algebraic group theory. The algebras $W_q^s(G)$ called q-W algebras are labeled
by (conjugacy classes of) elements $s$ of the Weyl group of $G$. The algebra
$W_q^s(G)$ is a quantization of a Poisson structure defined on the
corresponding transversal slice in $G$ with the help of Poisson reduction of a
Poisson bracket associated to a Poisson-Lie group $G^*$ dual to a
quasitriangular Poisson-Lie group. The algebras $W_q^s(G)$ can be regarded as
quantum group counterparts of W-algebras. However, in general they are not
deformations of the usual W-algebras.
