Polyploidy appears dependent on heterozygosity! The largest group of po1yp1oids, the a11opo1yp1oids (disomic po1yp1oids), have fixed heterozygosity in the two or more divergent genomes they possess (e.g., wheat, oats, cotton, tobacco, etc.). Hexaploid wheat, for example, although self-pollinated and basically homozygous at loci in each of its three genomes, has internal hybridity among loci with similar function in its three genomes. Disomic po1yp1oids thus are able to capitalize on the merits of both the se1f-and cross-fertilizing systems (1).The autopo1yp1oids (polysomic po1yp1oids) insure their heterozygosity through cross-pollination (e.g., alfalfa, birdsfoot trefoil, potato, and many grasses). We can find no example in crop plants of a successful polysomic polyploid species which is self-pollinated. Evidently the polysomic condition can not tolerate the homozygosity associated with self-pollination. The biochemical and physiological advantages of heterozygosity must be an important component of polyploid vigor (2). Polysomic po1yp1oids are represented by segregating and heterogeneous populations where maximum heterosis may be expressed by a few elite individuals in the population but not by the population per se. (Apomixis can preserve and perpetuate the elite polyploid individuals but will not be reviewed here.) In our seed reproduced polysomic po1yp1oids we can maximize the frequency of elite genotypes, but at present we cannot fix such genotypes in cultivated populations. As will be illustrated in this paper, maximum heterozygosity and heterosis does not occur in the Fl or single cross generation when parents are inbred (as it does 1n diploids or disomics) but occurs in the segregating double cross or an even later generation. Under polysomic polyploid conditions, the more inbred the parents, the lower the 471 W. H. Lewis (ed.), Polyploidy