Aim  Plants in islands have often evolved through adaptive radiation, providing the classical model of evolution of closely related species each with strikingly different morphological and ecological features and with low levels of genetic divergence. We emphasize the importance of an alternative (anagenetic) model of evolution, whereby a single island endemic evolves from a progenitor and slowly builds up genetic variation through time. Location  Continental and oceanic islands. Methods  We surveyed 2640 endemic angiosperm species in 13 island systems of the world, both oceanic and continental, for anagenetic and cladogenetic patterns of speciation. Genetic data were evaluated from a progenitor and derivative species pair in Ullung Island, Korea, and Japan. Results  We show that the anagenetic model of evolution is much more important in oceanic islands than previously believed, accounting for levels of endemic specific diversity from 7% in the Hawaiian Islands to 88% in Ullung Island, Korea, with a mean for all islands of 25%. Examination of an anagenetically derived endemic species in Ullung Island reveals genetic (amplified fragment length polymorphism) variation equal or nearly equal to that of its continental progenitor. Main conclusions  We hypothesize that, during anagenetic speciation, initial founder populations proliferate, and then accumulate genetic variation slowly through time by mutation and recombination in a relatively uniform environment, with drift and/or selection yielding genetic and morphological divergence sufficient for the recognition of new species. Low‐elevation islands with low habitat heterogeneity are highly correlated with high levels of anagenetic evolution, allowing prediction of levels of the two models of evolution from these data alone. Both anagenetic and adaptive radiation models of speciation are needed to explain the observed levels of specific and genetic diversity in oceanic islands.