Neural integration converts transient events into sustained neural activity. In the smooth pursuit eye movement system, neural integration is required to convert cerebellar output into the sustained discharge of extraocular motoneurons. We recorded the expression of integration in the time-varying firing rates of cerebellar and brainstem neurons in the monkey during pursuit of step-ramp target motion. Electrical stimulation with single shocks in the cerebellum identified brainstem neurons that are monosynaptic targets of inhibition from the cerebellar floccular complex. They discharge in relation to eye acceleration, eye velocity, and eye position, with a stronger acceleration signal than found in most other brainstem neurons. The acceleration and velocity signals can be accounted for by opponent contributions from the two sides of the cerebellum, without integration; the position signal implies participation in the integrator. Other neurons in the vestibular nucleus show a wide range of blends of signals related to eye velocity and eye position, reflecting different stages of integration. Neurons in the abducens nucleus discharge homogeneously in relation mainly to eye position, and reflect almost perfect integration of the cerebellar outputs. Average responses of neural populations and the diverse individual responses of large samples of individual neurons are reproduced by a hierarchical neural circuit based on a model suggested the anatomy and physiology of the larval zebrafish brainstem. The model uses a combination of feed forward and feedback connections to support a neural circuit basis for integration in monkeys and other species.