Speciation is often thought of as a slow process due to the waiting times for mutations that cause incompatibilities, and permit ecological differentiation or assortative mating. Cases of rapid speciation and particularly cases of rapid adaptive radiation into multiple sympatric species have remained somewhat mysterious. We review recent findings from speciation genomics that reveal an emerging commonality among such cases: reassembly of old genetic variation into new combinations facilitating rapid speciation and adaptive radiation. The polymorphisms in old variants frequently originated from hybridization at some point in the past. We discuss why old variants are particularly good fuel for rapid speciation, and hypothesize that variation in access to such old variants might contribute to the large variation in speciation rates observed in nature. Recent studies show that cases of rapid speciation and rapid species radiations often involve old genetic variants that arose long before the speciation events. Old genetic variation, previously tested by selection and occurring at higher allele frequency than new mutations, is a good substrate for speciation. Admixture variation from divergent lineages may be particularly important, potentially causing intrinsic and extrinsic incompatibilities, transgressive traits, or novel trait combinations in hybrid populations. We review the evidence for rapid speciation involving a ‘combinatorial mechanism’ – the reassembly of old genetic variants into novel combinations. This genetic mechanism might not only facilitate rapid speciation but also adaptive radiation and sympatric speciation, and it might contribute to variation in speciation rates among lineages.