The function of smooth pursuit is to keep the fovea pointed at a small visual target that moves smoothly across a patterned background. Chemical lesions, single cell recordings, and behavioral measures have shown that the cortical motion processing pathways form the afferent limb for pursuit. Important areas include at least the striate cortex and the middle temporal visual area, and probably the medial superior temporal visual area and the posterior parietal cortex. We argue that the visual inputs are transmitted through a simple sensory motor interface in the pons, to the efferent limb in the brain stem and cerebellum. The efferent limb uses neural velocity memory to maintain pursuit automatically. We present evidence that the velocity memory is provided, at least in part, by eye velocity positive feedback between the flocculus of the cerebellum and the brain stem. Finally, we use a computer model to show how the maintenance of pursuit can be simulated on a millisecond time scale. The structure and internal elements of the model are based on the biological experiments reviewed in our paper. In the past five years, progress on the neural basis of pursuit eye movements has been rapid. Several areas of research have made substantial contributions, by using combinations of new and conventional methods. Many of the pathways that contribute to pursuit have been identified, and their physiological activity and functions are becoming understood. Continuing progress promises to yield an understanding of one specific form of visually guided movement, at the level of neuronal circuits and behavior, in the primate.