The genus Avena consists of approximately 30 wild and cultivated oat species. Cultivated oat is an important food crop, yet the broader genetic diversity within the Avena gene pool remains underexplored and underexploited. Here, we characterize over 9000 wild and cultivated hexaploid oat accessions of global origin using genotyping-by-sequencing and explore population structure using multidimensional scaling and population-based clustering methods. We also conduct analyses to reveal chromosome regions associated with local adaptation, sometimes resulting from large-scale chromosome rearrangements. We report four distinct genetic populations within the wild species A. sterilis, a distinct population of cultivated A. byzantina, and multiple populations within cultivated A. sativa. Some chromosome regions associated with local adaptation are also associated with confirmed structural rearrangements on chromosomes 1A, 1C, 3C, 4C, and 7D. This work provides evidence suggesting multiple polyploid origins, multiple domestications, and/or reproductive barriers amongst Avena populations caused by differential chromosome structure.