The human visual system must perform complex visuospatial extrapolations (VSE) across space and time in order to extract shape and form from the retinal projection of a cluttered visual environment characterized by occluded surfaces and moving objects. Even if we exclude the temporal dimension, for instance when judging whether an extended finger is pointing towards one object or another, the mechanisms of VSE remain opaque. Here we investigated the neural correlates of VSE using functional magnetic resonance imaging in sixteen human observers while they judged the relative position of, or saccaded to, a (virtual) target defined by the extrapolated path of a pointer. Using whole brain and region of interest (ROI) analyses, we compared the brain activity evoked by these VSE tasks to similar control judgements or eye movements made to explicit (dot) targets that did not require extrapolation. The data show that activity in an occipitotemporal region that included the lateral occipital cortex (LOC) was significantly greater during VSE than during control tasks. A similar, though less pronounced, pattern was also evident in regions of the fronto-parietal cortex that included the frontal eye fields. However, none of the ROIs examined exhibited a significant interaction between target type (extrapolated/explicit) and response type (oculomotor/perceptual). These findings are consistent with a close association between visuoperceptual and oculomotor responses, and highlight a critical role for the LOC in the process of VSE.