Given the importance of variant SARS-CoV-2 viruses with altered receptor-binding or antigenic phenotypes, we sought to quantify the degree to which adaptive evolution is driving accumulation of mutations in the SARS-CoV-2 genome. Here we assessed adaptive evolution across genes in the SARS-CoV-2 genome by correlating clade growth with mutation accumulation as well as by comparing rates of nonsynonymous to synonymous divergence, clustering of mutations across the SARS-CoV-2 phylogeny and degree of convergent evolution of individual mutations. We find that spike S1 is the focus of adaptive evolution, but also identify positively-selected mutations in other genes that are sculpting the evolutionary trajectory of SARS-CoV-2. Adaptive changes in S1 accumulated rapidly, resulting in a remarkably high ratio of nonsynonymous to synonymous divergence that is 2.5X greater than that observed in HA1 at the beginning of the 2009 H1N1 pandemic.