Delineation of functional topography is critical to the evolving understanding of the cerebellum's role in a wide range of nervous system functions. We used data from the Human Connectome Project (n = 787) to analyze cerebellar fMRI task activation (motor, working memory, language, social and emotion processing) and resting-state functional connectivity calculated from cerebral cortical seeds corresponding to the peak Cohen's d of each task contrast. The combination of exceptional statistical power, activation from both motor and multiple non-motor tasks in the same participants, and convergent resting-state networks in the same participants revealed novel aspects of the functional topography of the human cerebellum. Consistent with prior studies there were two distinct representations of motor activation. Newly revealed were three distinct representations each for working memory, language, social, and emotional task processing that were largely separate for these four cognitive and affective domains. In most cases, the task-based activations and the corresponding resting-network correlations were congruent in identifying the two motor representations and the three non-motor representations that were unique to working memory, language, social cognition, and emotion. The definitive localization and characterization of distinct triple representations for cognition and emotion task processing in the cerebellum opens up new basic science questions as to why there are triple representations (what different functions are enabled by the different representations?) and new clinical questions (what are the differing consequences of lesions to the different representations?).