Here we report on the antibody and memory B cell responses of a cohort of 20 volunteers who received the Moderna (mRNA-1273) or Pfizer–BioNTech (BNT162b2) vaccine against SARS-CoV-2^{, , –}. Eight weeks after the second injection of vaccine, volunteers showed high levels of IgM and IgG anti-SARS-CoV-2 spike protein (S) and receptor-binding-domain (RBD) binding titre. Moreover, the plasma neutralizing activity and relative numbers of RBD-specific memory B cells of vaccinated volunteers were equivalent to those of individuals who had recovered from natural infection^,. However, activity against SARS-CoV-2 variants that encode E484K-, N501Y- or K417N/E484K/N501-mutant S was reduced by a small—but significant—margin. The monoclonal antibodies elicited by the vaccines potently neutralize SARS-CoV-2, and target a number of different RBD epitopes in common with monoclonal antibodies isolated from infected donors^{, , –}. However, neutralization by 14 of the 17 most-potent monoclonal antibodies that we tested was reduced or abolished by the K417N, E484K or N501Y mutation. Notably, these mutations were selected when we cultured recombinant vesicular stomatitis virus expressing SARS-CoV-2 S in the presence of the monoclonal antibodies elicited by the vaccines. Together, these results suggest that the monoclonal antibodies in clinical use should be tested against newly arising variants, and that mRNA vaccines may need to be updated periodically to avoid a potential loss of clinical efficacy.