Bread wheat (Triticum aestivum, genome BBAADD) is a young hexaploid species formed only 8500-9000 years ago through hybridization between a domesticated free-threshing tetraploid progenitor, genome BBAA, and Aegilops tauschii, the diploid donor of the D subgenome. Very soon after its formation, it spread globally from its cradle in the fertile crescent into new habitats and climates, to become a staple food of humanity. This extraordinary global expansion was probably enabled by allopolyploidy that accelerated genetic novelty through the acquisition of new traits, new intergenomic interactions, and buffering of mutations, and by the attractiveness of bread wheat’s large, tasty, and nutritious grain with high baking quality. New genome sequences suggest that the elusive donor of the B subgenome is a distinct (unknown or extinct) species rather than a mosaic genome. We discuss the origin of the diploid and tetraploid progenitors of bread wheat and the conflicting genetic and archaeological evidence on where it was formed and which species was its free-threshing tetraploid progenitor. Wheat experienced many environmental changes throughout its evolution, therefore, while it might adapt to current climatic changes, efforts are needed to better use and conserve the vast gene pool of wheat biodiversity on which our food security depends.