We provide a comprehensive account of the optical, electrical and mechanical properties that emerge from the self-assembly of colloidal nanocrystals or atomically precise nanoclusters into crystalline arrays with long-range order. We compare the correlation between the supercrystalline structure and these emergent properties with similar correlations in crystals of atoms to address the hypothesis that nanocrystals and nanoclusters exhibit quasi-atomic behaviour. We come to the conclusion that, effectively, this analogy is indeed justified, although the chemical origin for the same emergent properties are substantially different in crystals of atoms vs. supercrystals. We provide an outlook onto the most promising applications of supercrystals of nanocrystals and nanoclusters and discuss the challenges to be overcome before their commercialization.