  This paper presents a reduction procedure for nonholonomic systems admitting
suitable types of symmetries and conserved quantities. The full procedure
contains two steps. The first (simple) step results in a Chaplygin system,
described by an almost symplectic structure, carrying additional symmetries.
The focus of this paper is on the second step, which consists of a
Marsden-Weinstein--type reduction that generalizes constructions in [4,17]. The
almost symplectic manifolds obtained in the second step are proven to coincide
with the leaves of the reduced nonholonomic brackets defined in [7]. We
illustrate our construction with several classical examples.
