In Drosophila and humans, there are accumulating examples of loci with a significant excess of high-frequency-derived alleles or high levels of linkage disequilibrium, relative to a neutral model of a random-mating population of constant size. These are features expected after a recent selective sweep. Their prevalence suggests that positive directional selection may be widespread in both species. However, as I show here, these features do not persist long after the sweep ends: The high-frequency alleles drift to fixation and no longer contribute to polymorphism, while linkage disequilibrium is broken down by recombination. As a result, loci chosen without independent evidence of recent selection are not expected to exhibit either of these features, even if they have been affected by numerous sweeps in their genealogical history. How then can we explain the patterns in the data? One possibility is population structure, with unequal sampling from different subpopulations. Alternatively, positive selection may not operate as is commonly modeled. In particular, the rate of fixation of advantageous mutations may have increased in the recent past.