Solanum tuberosum, the cultivated potato of world commerce, is a primary food crop worldwide. Wild and cultivated potatoes form the germplasm base for international breeding efforts to improve potato in the face of a variety of disease, environmental and agronomic constraints. A series of national and international genebanks collect, characterize and distribute germplasm to stimulate and aid potato improvement. A knowledge of potato taxonomy and evolution guides collecting efforts, genebank operations and breeding. Past taxonomic treatments of wild and cultivated potato have differed tremendously among authors with regard to both the number of species recognized and the hypotheses of their interrelationships. In total, there are 494 epithets for wild and 626 epithets for cultivated taxa, including names not validly published. Recent classifications, however, recognize only about 100 wild species and four cultivated species. This paper compiles, for the first time, the epithets associated with all taxa of cultivated potato (many of which have appeared only in the Russian literature), places them in synonymy and provides lectotype designations for all names validly published where possible. We also summarize the history of differing taxonomic concepts in cultivated potato, and provide keys and descriptions for the four cultivated species. © 2011 The Linnean Society of London, Botanical Journal of the Linnean Society, 2011, 165, 107–155.