Scientists have pondered the perceptual effects of ocular motion, and those of its counterpart, ocular stillness, for over 200 years. The unremitting ‘trembling of the eye’ that occurs even during gaze fixation was first noted by Jurin in 1738. In 1794, Erasmus Darwin documented that gaze fixation produces perceptual fading, a phenomenon rediscovered in 1804 by Ignaz Paul Vital Troxler. Studies in the twentieth century established that Jurin's ‘eye trembling’ consisted of three main types of ‘fixational’ eye movements, now called microsaccades (or fixational saccades), drifts and tremor. Yet, owing to the constant and minute nature of these motions, the study of their perceptual and physiological consequences has met significant technological challenges. Studies starting in the 1950s and continuing in the present have attempted to study vision during retinal stabilization—a technique that consists on shifting any and all visual stimuli presented to the eye in such a way as to nullify all concurrent eye movements—providing a tantalizing glimpse of vision in the absence of change. No research to date has achieved perfect retinal stabilization, however, and so other work has devised substitute ways to counteract eye motion, such as by studying the perception of afterimages or of the entoptic images formed by retinal vessels, which are completely stable with respect to the eye. Yet other research has taken the alternative tack to control eye motion by behavioural instruction to fix one's gaze or to keep one's gaze still, during concurrent physiological and/or psychophysical measurements. Here, we review the existing data—from historical and contemporary studies that have aimed to nullify or minimize eye motion—on the perceptual and physiological consequences of perfect versus imperfect fixation. We also discuss the accuracy, quality and stability of ocular fixation, and the bottom–up and top–down influences that affect fixation behaviour. This article is part of the themed issue ‘Movement suppression: brain mechanisms for stopping and stillness’.