Smooth pursuit eye movements provide a model system for studying how visual inputs are transformed into commands for accurate movement. The neural circuit for pursuit eye movements is largely known and has strong parallels to the circuits for many other movements. Here, we outline progress in defining the conceptual operations that are performed by the pursuit circuit and in aligning those functions with neural circuit mechanisms. We discuss how the visual motion that drives pursuit is represented in the visual cortex, and how the visuomotor circuits decode that representation to estimate target direction and speed and to create motor commands. We outline a modulatory influence called gain control that evaluates the reliability and value of visual inputs and programs appropriate motor activity. Future research on pursuit in nonhuman primates holds the potential to reveal, at an unprecedented level of detail, how visuomotor circuits create coordinated actions.