Wheat is one of the major staple cereal food crops in India. However, most of the wheat-growing areas experience several biotic and abiotic stresses, resulting in poor quality grains and reduced yield. To ensure food security for the growing population in India, there is a compelling need to explore the untapped genetic diversity available in gene banks for the development of stress-resistant/tolerant cultivars. The improvement of any crop lies in exploring and harnessing the genetic diversity available in its genetic resources in the form of cultivated varieties, landraces, wild relatives, and related genera. A huge collection of wheat genetic resources is conserved in various gene banks across the globe. Molecular and phenotypic characterization followed by documentation of conserved genetic resources is a prerequisite for germplasm utilization in crop improvement. The National Genebank of India has an extensive and diverse collection of wheat germplasm, comprising Indian wheat landraces, primitive cultivars, breeding lines, and collection from other countries. The conserved germplasm can contribute immensely to the development of wheat cultivars with high levels of biotic and abiotic stress tolerance. Breeding wheat varieties that can give high yields under different stress environments has not made much headway due to high genotypes and environmental interaction, non-availability of truly resistant/tolerant germplasm, and non-availability of reliable markers linked with the QTL having a significant impact on resistance/tolerance. The development of new breeding technologies like genomic selection (GS), which takes into account the G × E interaction, will facilitate crop improvement through enhanced climate resilience, by combining biotic and abiotic stress resistance/tolerance and maximizing yield potential. In this review article, we have summarized different constraints being faced by Indian wheat-breeding programs, challenges in addressing biotic and abiotic stresses, and improving quality and nutrition. Efforts have been made to highlight the wealth of Indian wheat genetic resources available in our National Genebank and their evaluation for the identification of trait-specific germplasm. Promising genotypes to develop varieties of important targeted traits and the development of different genomics resources have also been highlighted.