  Various qualitative properties of solutions to the generalized Langevin
equation (GLE) in a periodic or a confining potential are studied in this
paper. We consider a class of quasi-Markovian GLEs, similar to the model that
was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem
(invariance principle), short time asymptotics and the white noise limit are
studied. Our proofs are based on a careful analysis of a hypoelliptic operator
which is the generator of an auxiliary Markov process. Systematic use of the
recently developed theory of hypocoercivity \cite{Vil04HPI} is made.
