Cerebellum means a little brain and much smaller that cerebral cortex in many animals.  In human the cerebellum has only 1/10 volume of cerebral cortex.  However, the cerebellum contains more neurons than cerebral cortex.  So, the cerebellum is one of the most densely neuron-populated area in the central nervous system. 
    The role of the cerebellum is described in patients in injury in the cerebellum such as gun-shot wound.   A wound caused by a bullet used to damage a large part of the brain.  However, the development of "rifle" gun allow a bullet to go straight and injury become significantly smaller in bullet's in and out of the body.  The wound to the cerebellum didn't cause the paralysis.  However, caused ataxia, a loss of the coordination of muscle movements.  The patient has difficulty in the coordinated movements of limbs such as clapping hands, walking, balancing the body.  Also have the problems in eye movements.  These patient have a problems in the test such as finger-to-nose test.  In addition, the patients with cerebellar injury has difficulties in non-declarative memories.  However, declarative memories are intact.
    The world-first neuronal plasticity that shows the "reduction" of neuronal transmission (long-term depression, LTD) is observed in cerebellum by Dr. Masao Ito of Tokyo University based on the mathematical models described by Albus and Marr.  Simple neuronal circuit of cerebellum and its clear role in motor coordination and motor learning attract researchers.  The molecular mechanisms underlying cerebellar LTD is well studied.