The Cerebellum is primarily connected to the motor cortex. Most think of this for the actuation of the body.
I should mention that it also modulates the connections from the motor and pre-motor projections to various other sections of the brain and allow the motor neurons to act as as drivers for action in the brain, prompting new states that are eventually perceived by the the temporal lobe as what we usually call “thoughts.”
The cerebellum facilitates the generation of longer and more stereotyped (Smoother?) sequences.
Some interesting papers on cortical-cerebellum connections:
I find this very interesting: From there, projections are sent to cerebral structures e.g. motor cortex. In turn, the cerebral cortex sends projections to pre-cerebellar nuclei, thereby forming multiple ‘cortico-cerebellar’ loops, reminiscent of cortico-basal ganglia loops
CEREBELLUM, BASAL GANGLIA AND CORTICAL CONNECTIONS UNMASKED IN HEALTH AND DISORDER
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5035040/
An important function of the cerebellum is controlling eye movements. A symptom of cerebellum damage is nystagmus.