Contrasting taxonomic treatments of potato landraces have continued over the last century, with the recognition of anywhere from 1 to 21 distinct Linnean species, or of Cultivar Groups within the single species Solanum tuberosum. We provide one of the largest molecular marker studies of any crop landraces to date, to include an extensive study of 742 landraces of all cultivated species (or Cultivar Groups) and 8 closely related wild species progenitors, with 50 nuclear simple sequence repeat (SSR) (also known as microsatellite) primer pairs and a plastid DNA deletion marker that distinguishes most lowland Chilean from upland Andean landraces. Neighbor-joining results highlight a tendency to separate three groups: (i) putative diploids, (ii) putative tetraploids, and (iii) the hybrid cultivated species S. ajanhuiri (diploid), S. juzepczukii (triploid), and S. curtilobum (pentaploid). However, there are many exceptions to grouping by ploidy. Strong statistical support occurs only for S. ajanhuiri, S. juzepczukii, and S. curtilobum. In combination with recent morphological analyses and an examination of the identification history of these collections, we support the reclassification of the cultivated potatoes into four species: (i) S. tuberosum, with two Cultivar Groups (Andigenum Group of upland Andean genotypes containing diploids, triploids, and tetraploids, and the Chilotanum Group of lowland tetraploid Chilean landraces); (ii) S. ajanhuiri (diploid); (iii) S. juzepczukii (triploid); and (iv) S. curtilobum (pentaploid). For other classifications, consistent and stable identifications are impossible, and their classification as species is artificial and only maintains the confusion of users of the gene banks and literature.