Leopards are the only big cats still widely distributed across the continents of Africa and Asia. They occur in a wide range of habitats and are often found in close proximity to humans. But despite their ubiquity, leopard phylogeography and population history have not yet been studied with genomic tools. Here, we present population-genomic data from 26 modern and historical samples encompassing the vast geographical distribution of this species. We find that Asian leopards are broadly monophyletic with respect to African leopards across almost their entire nuclear genomes. This profound genetic pattern persists despite the animals' high potential mobility, and despite evidence of transfer of African alleles into Middle Eastern and Central Asian leopard populations within the last 100,000 years. Our results further suggest that Asian leopards originated from a single out-of-Africa dispersal event 500-600 thousand years ago and are characterized by higher population structuring, stronger isolation by distance, and lower heterozygosity than African leopards. Taxonomic categories do not take into account the variability in depth of divergence among subspecies. The deep divergence between the African subspecies and Asian populations contrasts with the much shallower divergence among putative Asian subspecies. Reconciling genomic variation and taxonomy is likely to be a growing challenge in the genomics era.