The cerebellum has an important role in control and coordination of movements, but in some species, notably weakly electric fish of the family Mormyridae, anatomical, electrophysiological and behavioural evidence indicates that parts of cerebellar cortex are concerned with tracking movements of objects around the animal, rather than with controlling movements of the animal itself. The existence of such anomalies suggests that the cerebellum may not be exclusively, or even primarily, a structure for motor control. Evidence reviewed in this paper shows that the cerebellum is associated with sensory systems used for tracking movements of targets in the environment, as well as movements made by the animal itself, in all vertebrates, not just in a few isolated cases. The evidence indicates that the standard theory that the function of the cerebellum is control and coordination of movements only partially characterises cerebellar function. The cerebellum may be better characterised as a tracking system, with an important role in control and coordination of movements which arises because of an animal's need to track moving objects, to track its own movements, and to analyse the sensory consequences of movements in order to control movements. This theory not only predicts the known motor consequences of cerebellar dysfunction, it also predicts a specific kind of perceptual deficit caused by cerebellar dysfunction, namely an inability to accurately follow and predict trajectories of objects moving in the environment. A variety of behavioural and perceptual tasks in addition to motor control and movement tracking may require dynamical state estimation, and therefore may involve the cerebellum.