  In this paper we study the finitely generated algebras underlying $W$
algebras. These so called 'finite $W$ algebras' are constructed as Poisson
reductions of Kirillov Poisson structures on simple Lie algebras. The
inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$
into the simple Lie algebra in question. For arbitrary embeddings a coordinate
free formula for the reduced Poisson structure is derived. We also prove that
any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a
(semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that
generalized finite Toda systems are reductions of a system describing a free
particle moving on a group manifold and that they have finite $W$ symmetry. In
the second part we BRST quantize the finite $W$ algebras. The BRST cohomology
is calculated using a spectral sequence (which is different from the one used
by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in
one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part
of the paper we study the representation theory of finite $W$ algebras. It is
shown, using a quantum version of the generalized Miura transformation, that
the representations of finite $W$ algebras can be constructed from the
representations of a certain Lie subalgebra of the original simple Lie algebra.
As a byproduct of this we are able to construct the Fock realizations of
arbitrary finite $W$ algebras.
