Classic studies of the cerebellum before the middle of the twentieth century established the structural entity of the cerebellum and characterized its function as enabling animals and humans to carry out smooth and accurate movements, even at a high speed and without visual feedback. In the 1960s, neuronal circuit structures of the cerebellum were analyzed in detail, which promoted computational approaches toward the study of neuronal network principles of the cerebellum. In the 1970s and 1980s, vestibulo-ocular reflex adaptation, adaptive locomotion, eye blink conditioning, and learning in hand/arm movement were established as effective experimental paradigms for investigating neural mechanisms of cerebellar functions. In the 1980s, long-term depression (LTD) was discovered and considered as a memory process in the cerebellum; in the 1990s, complex signal transduction processes underlying LTD were revealed. It was also in the 1980s that computational approaches were advanced for modeling control system functions of the cerebellum. Currently, there are two alternative models proposed for VOR adaptation. In this decade, we envisage new developments toward the fusion of knowledge of the cerebellum at molecular and cellular levels and those in systems and computation. Studies of LTD will play a key role in pursuing this direction.