Perceptual sensitivity varies from moment to moment. One potential source of this variability is spontaneous fluctuations in cortical activity that can travel as waves1. Spontaneous travelling waves have been reported during anaesthesia2-7, but it is not known whether they have a role during waking perception. Here, using newly developed analytic techniques to characterize the moment-to-moment dynamics of noisy multielectrode data, we identify spontaneous waves of activity in the extrastriate visual cortex of awake, behaving marmosets (Callithrix jacchus). In monkeys trained to detect faint visual targets, the timing and position of spontaneous travelling waves before target onset predicted the magnitude of target-evoked activity and the likelihood of target detection. By contrast, spatially disorganized fluctuations of neural activity were much less predictive. These results reveal an important role for spontaneous travelling waves in sensory processing through the modulation of neural and perceptual sensitivity.