Drosophila mauritiana is an Indian Ocean island endemic species that diverged from its two sister species, Drosophila simulans and Drosophila sechellia, approximately 240,000 years ago. Multiple forms of incomplete reproductive isolation have evolved among these species, including sexual, gametic, ecological, and intrinsic postzygotic barriers, with crosses among all three species conforming to Haldane’s rule: F1 hybrid males are sterile and F1 hybrid females are fertile. Extensive genetic resources and the fertility of hybrid females have made D. mauritiana, in particular, an important model for speciation genetics. Analyses between D. mauritiana and both of its siblings have shown that the X chromosome makes a disproportionate contribution to hybrid male sterility. But why the X plays a special role in the evolution of hybrid sterility in these, and other, species remains an unsolved problem. To complement functional genetic analyses, we have investigated the population genomics of D. mauritiana, giving special attention to differences between the X and the autosomes. We present a de novo genome assembly of D. mauritiana annotated with RNAseq data and a whole-genome analysis of polymorphism and divergence from ten individuals. Our analyses show that, relative to the autosomes, the X chromosome has reduced nucleotide diversity but elevated nucleotide divergence; an excess of recurrent adaptive evolution at its protein-coding genes; an excess of recent, strong selective sweeps; and a large excess of satellite DNA. Interestingly, one of two centimorgan-scale selective sweeps on the D. mauritiana X chromosome spans a region containing two sex-ratio meiotic drive elements and a high concentration of satellite DNA. Furthermore, genes with roles in reproduction and chromosome biology are enriched among genes that have histories of recurrent adaptive protein evolution. Together, these genome-wide analyses suggest that genetic conflict and frequent positive natural selection on the X chromosome have shaped the molecular evolutionary history of D. mauritiana, refining our understanding of the possible causes of the large X-effect in speciation.