A remarkable challenge our brain must face constantly when interacting with the environment is represented by ambiguous and, at times, even missing sensory information. This is particularly compelling for visual information, being the main sensory system we rely upon to gather cues about the external world. It is not uncommon, for example, that objects catching our attention may disappear temporarily from view, occluded by visual obstacles in the foreground. Nevertheless, we are often able to keep our gaze on them throughout the occlusion or even catch them on the fly in the face of the transient lack of visual motion information. This implies that the brain can fill the gaps of missing sensory information by extrapolating the object motion through the occlusion. In recent years, much experimental evidence has been accumulated that both perceptual and motor processes exploit visual motion extrapolation mechanisms. Moreover, neurophysiological and neuroimaging studies have identified brain regions potentially involved in the predictive representation of the occluded target motion. Within this framework, ocular pursuit and manual interceptive behavior have proven to be useful experimental models for investigating visual extrapolation mechanisms. Studies in these fields have pointed out that visual motion extrapolation processes depend on manifold information related to short-term memory representations of the target motion before the occlusion, as well as to longer term representations derived from previous experience with the environment. We will review recent oculomotor and manual interception literature to provide up-to-date views on the neurophysiological underpinnings of visual motion extrapolation.