Wheat (Triticum aestivum L.) belonging to one of the most diverse and substantial families, Poaceae, is the principal cereal crop for the majority of the world’s population. This cereal is polyploidy in nature and domestically grown worldwide. Wheat is the source of approximately half of the food calories consumed worldwide and is rich in proteins (gluten), minerals (Cu, Mg, Zn, P, and Fe), vitamins (B-group and E), riboflavin, niacin, thiamine, and dietary fiber. Wheat seed-storage proteins represent an important source of food and energy and play a major role in the determination of bread-making quality. The two groups of wheat grain proteins, i.e., gliadins and glutenins, have been widely studied using SDS-PAGE and other techniques. Sustainable production with little input of chemicals along with high nutritional quality for its precise ultimate uses in the human diet are major focus areas for wheat improvement. An expansion in the hereditary base of wheat varieties must be considered in the wheat breeding program. It may be accomplished in several ways, such as the use of plant genetic resources, comprising wild relatives and landraces, germplasm-assisted breeding through advanced genomic tools, and the application of modern methods, such as genome editing. In this review, we critically focus on phytochemical composition, reproduction growth, types, quality, seed storage protein, and recent challenges in wheat breeding and discuss possible ways forward to combat those issues.