In this talk we will present some recent results on the long time asymptotics of the generalized Langevin equation (gLE). In particular, we study the ergodic
properties of the gLE and we also prove a homogenization (central limit) theorem. The analysis is based on the approximation of the gLE by a high (and possibly
infinite) dimensional degenerate Markovian system, and on the analysis of the spectrum of the generator of this Markov process.