Current wheat yield and consumption is considered in the context of the historical development of wheat, from early domestication through to modern plant breeding, the Green Revolution and wheat's place as one of the world's most productive and important crops in the 21st Century. The need for further improvement in the yield potential of wheat in order to meet current and impending challenges is discussed, including rising consumption and the demand for grain for fuel as well as food. Research on the complex genetics underlying wheat yield is described, including the identification of quantitative trait loci and individual genes, and the prospects of biotechnology playing a role in wheat improvement in the future are discussed. The challenge of preparing wheat to meet the problems of drought, high temperature and increasing carbon dioxide concentration that are anticipated to come about as a result of climate change is also reviewed. Wheat yield must be increased while not compromising food safety, and the emerging problem of processing contaminants is reviewed, focussing in particular on acrylamide, a contaminant that forms from free asparagine and reducing sugars during high temperature cooking and processing. Wheat breeders are strongly encouraged to consider the contaminant issue when breeding for yield.