1. We recorded the simple spike firing rate of gaze velocity Purkinje cells (GVP-cells) in the flocculus/ventral paraflocculus of two monkeys during the smooth pursuit eye movements evoked by a target that was initially at rest, started suddenly, moved at a constant velocity, and then stopped. 2. For target motion in the preferred direction, GVP-cells showed a large transient increase in firing rate at the onset of pursuit, a smaller but sustained increase during the maintenance of pursuit, and a smooth return to baseline firing with little undershoot at the offset of pursuit. For target motion in the nonpreferred direction, GVP-cells showed a small decrease in firing rate at the onset of pursuit, a similar sustained decrease during the maintenance of pursuit, but a large transient increase in firing rate at the offset of pursuit before returning to baseline firing. 3. We pooled the data in our sample of horizontal GVP-cells by subtracting the population average of firing rate recorded during pursuit in the nonpreferred direction from the population average recorded during pursuit in the preferred direction. We transformed this net population average by passing it through a model of the brain stem final common pathway and the oculomotor plant. This yielded a signal that closely matched the observed trajectory of eye velocity during pursuit. We conclude that the transient overshoots exhibited in the firing rate of GVP-cells can provide appropriate compensation for the lagging dynamics of the oculomotor plant.