  After some definitions, we review in the first part of this talk the
construction and classification of classical $W$ (super)algebras symmetries of
Toda theories. The second part deals with more recently obtained properties. At
first, we show that chains of $W$ algebras can be obtained by imposing
constraints on some $W$ generators: we call secondary reduction such a gauge
procedure on $W$ algebras. Then we emphasize the role of the Kac-Moody part,
when it exists, in a $W$ (super) algebra. Factorizing out this spin 1
subalgebra gives rise to a new $W$ structure which we interpret either as a
rational finitely generated $W$ algebra, or as a polynomial non linear
$W_\infty$ realization. (Plenary talk presented by P. SORBA at the $XXII^{th}$
International Conference on Differential Geometric Methods in Theoretical
Physics. Ixtapa Mexico, September 1993.)
