With the realization that much of the biological diversity on Earth has been generated by discrete evolutionary radiations, there has been a rapid increase in research into the biotic (key innovations) and abiotic (key environments) circumstances in which such radiations took place. Here we focus on the potential importance of population genetic structure and trait genetic architecture in explaining radiations. We propose a verbal model describing the stages of an evolutionary radiation: first invading a suitable adaptive zone and expanding both spatially and ecologically through this zone; secondly, diverging genetically into numerous distinct populations; and, finally, speciating. There are numerous examples of the first stage; the difficulty, however, is explaining how genetic diversification can take place from the establishment of a, presumably, genetically depauperate population in a new adaptive zone. We explore the potential roles of epigenetics and transposable elements (TEs), of neutral process such as genetic drift in combination with trait genetic architecture, of gene flow limitation through isolation by distance (IBD), isolation by ecology and isolation by colonization, the possible role of intra-specific competition, and that of admixture and hybridization in increasing the genetic diversity of the founding populations. We show that many of the predictions of this model are corroborated. Most radiations occur in complex adaptive zones, which facilitate the establishment of many small populations exposed to genetic drift and divergent selection. We also show that many radiations (especially those resulting from long-distance dispersal) were established by polyploid lineages, and that many radiating lineages have small genome sizes. However, there are several other predictions which are not (yet) possible to test: that epigenetics has played a role in radiations, that radiations occur more frequently in clades with small gene flow distances, or that the ancestors of radiations had large fundamental niches. At least some of these may be testable in the future as more genome and epigenome data become available. The implication of this model is that many radiations may be hard polytomies because the genetic divergence leading to speciation happens within a very short time, and that the divergence history may be further obscured by hybridization. Furthermore, it suggests that only lineages with the appropriate genetic architecture will be able to radiate, and that such a radiation will happen in a meta-population environment. Understanding the genetic architecture of a lineage may be an essential part of accounting for why some lineages radiate, and some do not.