Each time a locomoting fly turns, the visual image sweeps over the retina and generates a motion stimulus. Classic behavioral experiments suggested that flies use active neural-circuit mechanisms to suppress the perception of self-generated visual motion during intended turns. Direct electrophysiological evidence, however, has been lacking. We found that visual neurons in Drosophila receive motor-related inputs during rapid flight turns. These inputs arrived with a sign and latency appropriate for suppressing each targeted cell's visual response to the turn. Precise measurements of behavioral and neuronal response latencies supported the idea that motor-related inputs to optic flow-processing cells represent internal predictions of the expected visual drive induced by voluntary turns. Motor-related inputs to small object-selective visual neurons could reflect either proprioceptive feedback from the turn or internally generated signals. Our results in Drosophila echo the suppression of visual perception during rapid eye movements in primates, demonstrating common functional principles of sensorimotor processing across phyla.