We analyzed gaze shifts made by trained rhesus monkeys with completely unrestrained heads during performance of a delayed gaze shift task. Subjects made horizontal, vertical, and oblique gaze shifts to visual targets. We found that coordinated eye-head movements are characterized by a set of lawful relationships, and that the initial position of the eyes in the orbits and the direction of the gaze shift are two factors that influence these relationships. Head movements did not contribute to the change in gaze position during small gaze shifts (<20 degrees) directed along the horizontal meridian, when the eyes were initially centered in the orbits. For larger gaze shifts (25-90 degrees), the head contribution to the gaze shift increased linearly with increasing gaze shift amplitude, and eye movement amplitude saturated at an asymptotic amplitude of approximately 35 degrees. When the eyes began deviated in the orbits contralateral to the direction of the ensuing gaze shift, the head contributed less and the eyes more to amplitude-matched gaze shifts. The relative timing of eye and head movements was altered by initial eye position; head latency relative to gaze onset increased as the eyes began in more contralateral initial positions. The direction of the gaze shift also affected the relative amplitudes of eye and head movements; as gaze shifts were made in progressively more vertical directions, eye amplitude increased and head contribution declined systematically. Eye velocity was a saturating function of gaze amplitude for movements without a head contribution (gaze amplitude <20 degrees). As head contribution increased with increasing gaze amplitude (20-60 degrees), peak eye velocity declined by >200 degrees/s and head velocity increased by 100 degrees/s. For constant-amplitude eye movements (approximately 30 degrees), eye velocity declined as the velocity of the concurrent head movement increased. On the basis of these relationships, it is possible to accurately predict gaze amplitude, the amplitudes of the eye and head components of the gaze shift, and gaze, eye, and head velocities, durations and latencies if the two-dimensional displacement of the target and the initial position of the eyes in the orbits are known. These data indicate that signals related to the initial positions of the eyes in the orbits and the direction of the gaze shift influence separate eye and head movement commands. The hypothesis that this divergence of eye and head commands occurs downstream from the superior colliculus is supported by recent electrical stimulation and single-unit recording data.