The platypus has a remarkable combination of mammalian and reptilian characteristics and, as an egg-laying mammal, alongside the echidna, it occupies a unique place in the phylogenetic tree. Despite widespread interest in its unusual biology, much remains to be learned about its dispersal patterns, population structure, and recent evolutionary history. To address this, we sequenced the genomes of 57 platypuses from across the species’ range (eastern mainland Australia and Tasmania). Our results show very strong population structure, with our sampling locations corresponding to discrete populations between which there is no evidence for recent gene flow. We found that 31 of the 57 samples had at least a third-degree relative amongst other samples from the same river system, indicating that it is not uncommon for related individuals to remain in the same stretch of stream. Despite this, we see many individuals with little evidence of inbreeding, suggesting biological mechanisms to avoid mating with close relatives. Data from a family quartet allowed us to estimate the de novo mutation rate in the platypus at 1.2 - 9.6 × 10-8 bp/generation, one of the first direct estimates made in a non-model organism. Some patterns of similarity and differences between populations are not easy to reconcile with geography, suggesting historical migration patterns more complicated than predicted by simple isolation-by-distance models. Estimates of historical effective population sizes showed that Queensland populations underwent a strong bottleneck likely during the Last Glacial Maximum, when there is evidence for bottlenecks in other rainforest species in that region. Only the population in the Wet Tropics recovered, and the populations in central Queensland appear to be priorities for conservation due to small effective population size and low diversity. This study demonstrates the power of whole-genome re-sequencing of natural populations of an evolutionarily important species with a problematic reference genome