The nervous systems of most vertebrates include both the cerebellum and structures that are architecturally similar to the cerebellum. The cerebellum-like structures are sensory structures that receive input from the periphery in their deep layers and parallel fiber input in their molecular layers. This review describes these cerebellum-like structures and compares them with the cerebellum itself. The cerebellum-like structures in three groups of fish act as adaptive sensory processors in which the signals conveyed by parallel fibers in the molecular layer predict the patterns of sensory input to the deep layers through a process of associative synaptic plasticity. Similarities between the cerebellum-like structures and the cerebellum suggest that the cerebellum may also generate predictions about expected sensory inputs or states of the system, as suggested also by clinical, experimental, and theoretical studies of the cerebellum. Understanding the process of predicting sensory patterns in cerebellum-like structures may therefore be a source of insight into cerebellar function.