Smooth-pursuit eye movements are used to stabilize the image of a moving object of interest on the fovea, thus guaranteeing its high-acuity scrutiny. Such movements are based on a phylogenetically recent cerebro-ponto-cerebellar pathway that has evolved in parallel with foveal vision. Recent work has shown that a network of several cerebrocortical areas directs attention to objects of interest moving in three dimensions and reconstructs the trajectory of the target in extrapersonal space, thereby integrating various sources of multimodal sensory and efference copy information, as well as cognitive influences such as prediction. This cortical network is the starting point of a set of parallel cerebrofugal projections that use different parts of the dorsal pontine nuclei and the neighboring rostral nucleus reticularis tegmenti pontis as intermediate stations to feed two areas of the cerebellum, the flocculus-paraflocculus and the posterior vermis, which make mainly complementary contributions to the control of smooth pursuit.