There are three systems in cerebellum:
1.Cortex to pons to deep cerebellar nuclei via mossy fiber projections
2. Neocortex to dentate nucleus to thalamus to cortex
3. Olive to cerebellum
What are they for? Cerebellum allows organism to predict or anticipate motor behaviors and produce rapi automatic predictive behaviors. These are called "internal models" and allow the cerebellum to copy the contents of the cortical "working memory" or what the brain intends to do, which forms the basis of cerebellar cognitive control model. The cerebellum can also adapt behavior across similar contexts, and become increasingly precise with practice.
Overlearned motor behaviors are represented in the anterior lobes of the cerebellum, whereas learning new sensorimotor information occurs in the posterior and inferior lobes of the cerebellum.
Koziol et al argue that it does for thought what it does for motor function and teaches the prefrontal cortex the anticipated rewards of an an action or thought.It underpins thinking and anticipatory cognitive control.
There are segregated modularities in the posterior lateral cerebellum for different tools.