The unique inheritance pattern of the X chromosome exposes it to natural selection in a way that is different from that of the autosomes, potentially resulting in accelerated evolution. We perform a comparative analysis of X chromosome polymorphism in 10 great ape species, including humans. In most species, we identify striking megabase-wide regions, where nucleotide diversity is less than 20% of the chromosomal average. Such regions are found exclusively on the X chromosome. The regions overlap partially among species, suggesting that the underlying targets are partly shared among species. The regions have higher proportions of singleton SNPs, higher levels of population differentiation, and a higher nonsynonymous-to-synonymous substitution ratio than the rest of the X chromosome. We show that the extent to which diversity is reduced is incompatible with direct selection or the action of background selection and soft selective sweeps alone, and therefore, we suggest that very strong selective sweeps have independently targeted these specific regions in several species. The only genomic feature that we can identify as strongly associated with loss of diversity is the location of testis-expressed ampliconic genes, which also have reduced diversity around them. We hypothesize that these genes may be responsible for selective sweeps in the form of meiotic drive caused by an intragenomic conflict in male meiosis.