The wheat group has evolved through allopolyploidization, namely, through hybridization among species from the plant genera Aegilops and Triticum followed by genome doubling. This speciation process has been associated with ecogeographical expansion and with domestication. In the past few decades, we have searched for explanations for this impressive success. Our studies attempted to probe the bases for the wide genetic variation characterizing these species, which accounts for their great adaptability and colonizing ability. Central to our work was the investigation of how allopolyploidization alters genome structure and expression. We found in wheat that allopolyploidy accelerated genome evolution in two ways: (1) it triggered rapid genome alterations through the instantaneous generation of a variety of cardinal genetic and epigenetic changes (which we termed "revolutionary" changes), and (2) it facilitated sporadic genomic changes throughout the species' evolution (i.e., evolutionary changes), which are not attainable at the diploid level. Our major findings in natural and synthetic allopolyploid wheat indicate that these alterations have led to the cytological and genetic diploidization of the allopolyploids. These genetic and epigenetic changes reflect the dynamic structural and functional plasticity of the allopolyploid wheat genome. The significance of this plasticity for the successful establishment of wheat allopolyploids, in nature and under domestication, is discussed.