How our vision remains stable in spite of the interruptions produced by saccadic eye movements has been a repeatedly revisited perceptual puzzle. The major hypothesis is that a corollary discharge (CD) or efference copy signal provides information that the eye has moved, and this information is used to compensate for the motion. There has been progress in the search for neuronal correlates of such a CD in the monkey brain, the best animal model of the human visual system. In this article, we briefly summarize the evidence for a CD pathway to frontal cortex, and then consider four questions on the relation of neuronal mechanisms in the monkey brain to stable visual perception. First, how can we determine whether the neuronal activity is related to stable visual perception? Second, is the activity a possible neuronal correlate of the proposed transsaccadic memory hypothesis of visual stability? Third, are the neuronal mechanisms modified by visual attention and does our perceived visual stability actually result from neuronal mechanisms related primarily to the central visual field? Fourth, does the pathway from superior colliculus through the pulvinar nucleus to visual cortex contribute to visual stability through suppression of the visual blur produced by saccades?