Significance Understanding the ongoing evolution of SARS-CoV-2 is essential to control and ultimately end the pandemic. We analyzed more than 300,000 SARS-CoV-2 genomes available as of January 2021 and demonstrate adaptive evolution of the virus that affects, primarily, multiple sites in the spike and nucleocapsid protein. Selection appears to act on combinations of mutations in these and other SARS-CoV-2 genes. Evolution of the virus is accompanied by ongoing adaptive diversification within and between geographic regions. This diversification could substantially prolong the pandemic and the vaccination campaign, in which variant-specific vaccines are likely to be required.