The past 25 years have seen the functional domain of the cerebellum extend beyond the realm of motor control, with considerable discussion of how this subcortical structure contributes to cognitive domains including attention, memory, and language. Drawing on evidence from neuroanatomy, physiology, neuropsychology, and computational work, sophisticated models have been developed to describe cerebellar function in sensorimotor control and learning. In contrast, mechanistic accounts of how the cerebellum contributes to cognition have remained elusive. Inspired by the homogeneous cerebellar microanatomy and a desire for parsimony, many researchers have sought to extend mechanistic ideas from motor control to cognition. One influential hypothesis centers on the idea that the cerebellum implements internal models, representations of the context‐specific dynamics of an agent's interactions with the environment, enabling predictive control. We briefly review cerebellar anatomy and physiology, to review the internal model hypothesis as applied in the motor domain, before turning to extensions of these ideas in the linguistic domain, focusing on speech perception and semantic processing. While recent findings are consistent with this computational generalization, they also raise challenging questions regarding the nature of cerebellar learning, and may thus inspire revisions of our views on the role of the cerebellum in sensorimotor control.