The opportunities of molecular genetic approaches to manage the wheat grain’s technological properties, influencing the quality of end products of bread industry, were considered within this review. At present, the traditional range is crowded out with an increase in grain production, the quality of the mass varieties of bread is deteriorating, and tens of different substances of biological and chemical origin are used as improvers in bread baking. Meanwhile, wheat’s genetic potential allows us to create varieties for the production of grain with technological characteristics suitable for high-quality bread production. Multiple examples of the creation of varieties for the production of 1st and 2nd class grain exist in the history of domestic breeding; modern molecular genetics suggest approaches that in combination with traditional breeding methods enable the accelerated creation of new varieties adapted to the conditions and requirements of the baking industry by using the natural genetic potential of wheat. The authors summarized the data on the diversity of the requirements imposed on the grain and flour to be eventually used in different foods. The statistical data on the volume and structure of the grain quality in Russia from 2011 to 2014 were analyzed. A significant deformation of the quality structure of the produced wheat grain in favor of less valuable classes was detected. A short retrospective analysis of the studies in the field of wheat genetics that demonstrates the role of genetic factors in the development of technological properties of grain and flour was conducted. Different approaches to accelerated breeding of varieties with specified properties for the prospect of developing studies in the field of plant molecular genetics were considered. The examples illustrating the possibility and expedience of using DNA diagnostics methods at different stages of the process, during which the genetic potential (inherent in the gene pool of food crops) is realized and affects the final product quality, are given. The results of molecular genetic studies on determining the localization and structure of the genes involved in developing the technological properties of the wheat grain (the content of protein and wet gluten, milling properties, rheological properties of the flour and dough, flour color, and starch properties) were considered. The data on diagnostic DNA markers, suitable for the efficient selection of genotypes instead of the time-consuming analysis of technological characteristics at intermediate breeding stages, were summarized. Thus, the information on bread wheat’s genetic potential and modern technological approaches, which constitute the basis for a change of the direction from increased chemicalization towards a softer and more organic effect on the quality of the main raw material and products in the entire “grain–flour—bread” food chain, is summarized.