While classical views of cerebellar learning have suggested that this structure predominantly operates according to an error-based supervised learning rule to refine movements, emerging evidence suggests that the cerebellum may also harness a wider range of learning rules to contribute to a variety of behaviors, including cognitive processes. Together, such evidence points to a broad role for cerebellar circuits in generating and testing predictions about movement, reward, and other non-motor operations. However, this expanded view of cerebellar processing also raises many new questions about how such apparent diversity of function arises from a structure with striking homogeneity. Hence, this review will highlight both current evidence for predictive cerebellar circuit function that extends beyond the classical view of error-driven supervised learning, as well as open questions that must be addressed to unify our understanding cerebellar circuit function.