A number of major crop species, such as allohexaploid wheat and autotetraploid potato are polyploid. Potato is the fourth most important crop in terms of production and has become an important food source in many countries. Therefore, the molecular analysis was directed towards investigating ways to develop markers to assist the potato breeding process; for example breeding for powdery scab disease resistance, and tolerance to cold induced sweetening. Polyploids have more possible genotypes per population, allele dosage effects and increased marker complexity compared to diploids. Potato is also outcrossing and therefore highly heterozygous. Various methods for detecting marker-trait associations including, linkage, quantitative trait locus (QTL) and association mapping were studied and protocols developed. A mapping population was produced and a number of traits were measured including powdery scab resistance. Powdery scab disease assays were carried out over six seasons and markers associated with disease resistance were identified. Markers associated with resistance to powdery scab were identified on chromosomes I, IV, V, VI, VIII and IX using analysis of variance (ANOVA). Linkage maps were produced for each parent of the population and QTL associated with resistance and susceptibility to disease were identified using interval mapping, which revealed QTL on chromosomes II, V, VII , VIII, IX and an unanchored linkage group. QTL were detected across years on regions of chromosomes VIII and IX. These QTL results had some overlap with the marker-trait associations that were identified using ANOVA analysis. Another marker identification technique was tested, known as association or linkage disequilibrium mapping. Alleles of candidate genes were tested for association with cold-induced sweetening using a germplasm collection. The alleles identified as important were of the apoplastic invertase and UGPase genes and a unique interaction between alleles of the apoplastic invertase and apoplastic invertase inhibitor was also detected. This thesis describes the first study into the genetics of powdery scab resistance and the markers identified as associated with resistance will be validated for use in a marker-assisted selection (MAS) programme. The tools and resources developed as part of this thesis are vital to the potato breeding programme that requires the identification of associated molecular markers.