Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused millions of deaths and substantial morbidity worldwide. Intense scientific effort to understand the biology of SARS-CoV-2 has resulted in daunting numbers of genomic sequences. We witnessed evolutionary events that could mostly be inferred indirectly before, such as the emergence of variants with distinct phenotypes, for example transmissibility, severity and immune evasion. This Review explores the mechanisms that generate genetic variation in SARS-CoV-2, underlying the within-host and population-level processes that underpin these events. We examine the selective forces that likely drove the evolution of higher transmissibility and, in some cases, higher severity during the first year of the pandemic and the role of antigenic evolution during the second and third years, together with the implications of immune escape and reinfections, and the increasing evidence for and potential relevance of recombination. In order to understand how major lineages, such as variants of concern (VOCs), are generated, we contrast the evidence for the chronic infection model underlying the emergence of VOCs with the possibility of an animal reservoir playing a role in SARS-CoV-2 evolution, and conclude that the former is more likely. We evaluate uncertainties and outline scenarios for the possible future evolutionary trajectories of SARS-CoV-2.