Aim Central America is largely covered by hyperdiverse, yet poorly understood, rain forests. Understorey palms are diverse components of these forests, but little is known about their historical assembly. It is not clear when palms in Central America reached present diversity levels and whether most species arrived from neighbouring regions or evolved locally. We addressed these questions using the most species rich American palm clades indicative of rain forests. We reconstructed and compared their phylogenomic and biogeographical history with the diversification of 54 other plant lineages, to gain a better understanding of the processes that shaped the assembly of Central American rain forests. Location Central America. Time period Cretaceous to present. Major taxa studied Arecaceae,Arecoideae,Bactridinae, Chamaedoreeae, Geonomateae. Methods We sampled 218 species through fieldwork and living collections. We sequenced their genomic DNA using target sequence capture procedures. Using 12 calibration points, we reconstructed dated phylogenies under three approaches (multispecies coalescent, maximum likelihood and Bayesian inference), conducted biogeographical analyses (dispersal extinction cladogenesis) and estimated phylogenetic diversity metrics. Results Dated phylogenies revealed intense diversification in Central America from 12 Ma. Local diversification events were four times more frequent than dispersal events, and we found strong phylogenetic clustering in relationship to Central America. Main conclusions Our results suggest that most understorey palm species that characterize the Central American rain forests today evolved locally after repeated dispersal events, mostly from South America. Understorey palms in Central American rain forests diversified primarily after closure of the Central American Seaway at c. 13 Ma, suggesting that the Great American Biotic Interchange was a major trigger for plant diversification in Central American rain forests. This recent diversification contrasts with the much earlier existence of rain forest palms in neighbouring South America since c. 58 Ma. We found similar timings of diversification in 54 other seed plant lineages, suggesting an unexpectedly recent assembly of the hyperdiverse Central American flora.