Theory predicts that in allopatric populations, genomic divergence and reproductive barriers may be driven by random genetic drift and thereby evolve slowly in large populations. However, local adaptation and divergence under selection may also play important roles, which remain poorly characterised. Here, we address three key questions in young allopatric species: (a) How widespread are genomic signatures of adaptive divergence?, (b) What is the functional space along which young sister species show divergence at the genomic level? and (c) How quickly might prezygotic and postzygotic reproductive barriers evolve? Analysis of 82 re-sequenced genomes of the Oriental Papilio polytes species group revealed surprisingly widespread hotspots of intense selection and selective sweeps at hundreds of genes, spanning all chromosomes, rather than divergence only in a few genomic islands. These genes are involved in diverse ecologically important adaptive functions such as wing development, colour patterning, courtship behaviour, mimicry, pheromone synthesis and olfaction, and host plant use and digestion of secondary metabolites, that could contribute to local adaptation and subsequent reproductive isolation. Divergence at such functional genes appeared to have evolved in conjunction with reproductive consequences: behavioural and hybridisation experiments revealed strong assortative mate preference (prezygotic barriers) as well as postzygotic barriers to hybridisation in timespans as short as 1.5 my, indicating that speciation was already complete rather than incipient. Our study thus demonstrates an underappreciated role of intense selection and potential local adaptation in creating genome-wide hotspots of rapid molecular evolution and divergence during differentiation and speciation in young allopatric species.