Bread wheat (Triticum aestivum) is one of the world's most important crops; however, a low level of genetic diversity within commercial breeding accessions can significantly limit breeding potential. In contrast, wheat relatives exhibit considerable genetic variation and so potentially provide a valuable source of novel alleles for use in breeding new cultivars. Historically, gene flow between wheat and its relatives may have contributed novel alleles to the bread wheat pangenome. To assess the contribution made by wheat relatives to genetic diversity in bread wheat, we used markers based on single nucleotide polymorphisms to compare bread wheat accessions, created in the past 150 years, with 45 related species. We show that many bread wheat accessions share near-identical haplotype blocks with close relatives of wheat's diploid and tetraploid progenitors, while some show evidence of introgressions from more distant species and structural variation between accessions. Hence, introgressions and chromosomal rearrangements appear to have made a major contribution to genetic diversity in cultivar collections. As gene flow from relatives to bread wheat is an ongoing process, we assess the impact that introgressions might have on future breeding strategies.