  Some aspects of the geometry and the dynamics of generalized Chaplygin
systems are investigated. First, two different but complementary approaches to
the construction of the reduced dynamics are reviewed: a symplectic approach
and an approach based on the theory of affine connections. Both are mutually
compared and further completed. Next, a necessary and sufficient condition is
derived for the existence of an invariant measure for the reduced dynamics of
generalized Chaplygin systems of mechanical type. A simple example is then
constructed of a generalized Chaplygin system which does not verify this
condition, thereby answering in the negative a question raised by Koiller.
