Transposable elements (TEs) are increasingly being recognized as powerful facilitators of evolution. We propose the TE-Thrust hypothesis to encompass TE-facilitated processes by which genomes self-engineer coding, regulatory, karyotypic or other genetic changes. Although TEs are occasionally harmful to some individuals, genomic dynamism caused by TEs can be very beneficial to lineages. This can result in differential survival and differential fecundity of lineages. Lineages with an abundant and suitable repertoire of TEs have enhanced evolutionary potential and, if all else is equal, tend to be fecund, resulting in species-rich adaptive radiations, and/or they tend to undergo major evolutionary transitions. Many other mechanisms of genomic change are also important in evolution, and whether the evolutionary potential of TE-Thrust is realized is heavily dependent on environmental and ecological factors. The large contribution of TEs to evolutionary innovation is particularly well documented in the primate lineage. In this paper, we review numerous cases of beneficial TE-caused modifications to the genomes of higher primates, which strongly support our TE-Thrust hypothesis.