A fundamental problem in the generation of goal-directed behaviour is caused by the inevitable latency of biological sensory systems. Behaviour which is fully synchronised with the triggering sensory event can only be executed if the occurrence of this event can be predicted based on prior information. Smooth-pursuit eye movements are a classical and well-established example of goal-directed behaviour. The execution of these eye movements is thought to be very closely linked to the processing of visual motion signals. Here, we show that healthy human subjects as well as trained rhesus monkeys are able to initiate smooth-pursuit eye movements in anticipation of a moving target. These anticipatory pursuit eye movements are scaled to the velocity of the expected target. Furthermore, we can exclude the possibility that anticipatory pursuit is simply an after-pursuit of the previous trial. Visually-guided pursuit is only marginally affected by the presence of a structured background. However, the presence of a structured background severely impedes the ability to perform anticipatory pursuit. More generally, our data provide additional evidence that the cognitive oculomotor repertoires of human and monkeys are similar, at least with respect of smooth-pursuit in the prediction of an appearing target.