How does the nervous system guide the muscular periphery during the acquisition of a new motor skill? This is a fundamental question for researchers trying to understand the neural basis of motor learning. Recent advances in studying a valuable example of short-term motor learning, namely the adaptation of saccadic eye movements, have revealed neuronal processes in the cerebellum that underlie the unfolding of the learned behavior. In this review, we describe the latest findings from electrophysiology studies of saccadic adaptation and how they can generalize to more elaborate examples of cerebellum-dependent adaptation of movements. We focus our discussion on the plastic changes that are observed in the firing properties of Purkinje cells during the acquisition of the wanted motor response and describe how the altered activity of these neurons modifies the dynamics of the cerebellar microcircuitry. We finally demonstrate how such task-related modifications in the cerebellum are appropriate to fine-tune extracerebellar pre-motor structures and induce the learned behavior.