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An impressive body of research over the past 35 years has implicated the human cerebellum in a broad range of functions beyond motor control, including language, working memory, cognitive control, and social cognition. The relatively uniform anatomy and physiology of the cerebellar cortex has given rise to the universal cerebellar transform hypothesis (UCT), the idea that the cerebellum can be conceptualized as a module providing a basic computation that is exploited across diverse domains. Proposed UCTs focus on the concepts of prediction and coordination.  To make these ideas computationally meaningful, we need to specify the constraints on cerebellar processing: What are the types of prediction supported by the cerebellum and what do we mean when speaking of “mental coordination”?  I will address these questions in two parts.  First, using variants of sensorimotor adaptation tasks, I will describe properties of a cerebellar-dependent learning process that ensure our movements remain exquisitely calibrated.  Second, I will describe how the results have motivated a new hypothesis concerning how the cerebellum might contribute to cognition, focusing on its role in supporting dynamic mental transformations.