Breeding of arable crops such as wheat has led to substantial improvements in yield, quality of produce, traits of agronomic value and resistance to disease and abiotic stress. Resistance to many diseases and pests of wheat has improved through a combination of innovation by breeders, driven by competition, and independent assessment of cultivars in Recommended List trials. In addition to pleiotropic effects of resistance genes, two further limitations on breeding for disease resistance are linkage drag and competition drag. Linkage drag, the slow rate of decline of linkage disequilibrium between closely linked genes, can be especially significant in wheat, particularly after the introgression of chromosome segments from related wild grasses. For example it slowed the deployment of the Pch1 gene for resistance to eyespot in leading cultivars because Pch1 is on a chromosome segment that also reduces yield. Linkage drag also inhibited the use of Pm16 for resistance to powdery mildew. Competition drag can be described as the delay in the deployment of a useful gene because of the additional time needed to raise yield and quality to the standard of the current market-leading cultivars. This is exemplified by the near-absence of resistance to Soil-borne cereal mosaic virus in most modern varieties, despite no evidence for detrimental pleiotropic effects. Changes in agricultural practice and regulation of pesticides are increasing demand for durable resistance to disease but wherever possible, improvement of resistance should not constrain selection for yield which remains the most significant trait in leading wheat cultivars for the UK.