The principal cultivated potato (Solanum tuberosum) has mainly been vegetatively propagated through its tubers. Potato breeders have therefore made planned artificial hybridizations to generate genetically unique seedlings and their clonal descendants from which to select new cultivars for tuber propagation. After the initial hybridizations, no more sexual reproduction was required to produce a successful new cultivar, which depended on choosing the correct breeding objectives and the ability to recognize a clone that met those objectives. Any impact of the new science of genetics after 1900 needed to be through the production of parental material of known genetic constitution and predictable offspring. This included making use of the many wild tuber-bearing relatives of the potato in Central and South America, as well as the abundance of landraces in South America. This review looks at the history of how potato geneticists: 1) established that the principal cultivated potato is a tetraploid that displays tetrasomic inheritance (2n = 4x = 48); 2) developed progeny tests to determine the dosage of major genes for qualitative traits in potential parents, and also progeny tests for their general combining abilities for quantitative traits; and 3) provided molecular markers for the marker assisted selection of major genes and quantitative trait alleles of large effect, and for the genomic selection of many alleles of small effect. It is argued that the concepts of population genetics are required by breeders, once a number of cycles of hybridization and cultivar production are considered for the genetic improvement of potato crops.