Visual tracking of objects in a noisy environment is a difficult problem that has been solved by the primate oculomotor system, but remains unsolved in robotics. In primates, smooth pursuit eye movements match eye motion to target motion to keep the eye pointed at smoothly moving targets. We have used computer models as a tool to investigate possible computational strategies underlying this behavior. Here, we present a model based upon behavioral data from monkeys. The model emphasizes the variety of visual signals available for pursuit and, in particular, includes a sensitivity to the acceleration of retinal images. The model was designed to replicate the initial eye velocity response observed during pursuit of different target motions. The strength of the model is that it also exhibits a number of emergent properties that are seen in the behavior of both humans and monkeys. This suggests that the elements in the model capture important aspects of the mechanism of visual tracking by the primate smooth pursuit system.