The role of positive selection in human evolution remains controversial. On the one hand, scans for positive selection have identified hundreds of candidate loci and the genome-wide patterns of polymorphism show signatures consistent with frequent positive selection. On the other hand, recent studies have argued that many of the candidate loci are false positives and that most apparent genome-wide signatures of adaptation are in fact due to reduction of neutral diversity by linked recurrent deleterious mutations, known as background selection. Here we analyze human polymorphism data from the 1,000 Genomes project (Abecasis et al. 2012) and detect signatures of pervasive positive selection once we correct for the effects of background selection. We show that levels of neutral polymorphism are lower near amino acid substitutions, with the strongest reduction observed specifically near functionally consequential amino acid substitutions. Furthermore, amino acid substitutions are associated with signatures of recent adaptation that should not be generated by background selection, such as the presence of unusually long and frequent haplotypes and specific distortions in the site frequency spectrum. We use forward simulations to show that the observed signatures require a high rate of strongly adaptive substitutions in the vicinity of the amino acid changes. We further demonstrate that the observed signatures of positive selection correlate more strongly with the presence of regulatory sequences, as predicted by ENCODE (Gerstein et al. 2012), than the positions of amino acid substitutions. Our results establish that adaptation was frequent in human evolution and provide support for the hypothesis of King and Wilson (King and Wilson 1975) that adaptive divergence is primarily driven by regulatory changes.