The COVID-19 pandemic has had widespread effects across the globe, and its causative agent, SARS-CoV-2, continues to spread. Effective interventions need to be developed to end this pandemic. Single and combination therapies with monoclonal antibodies have received emergency use authorization, and more treatments are under development. Furthermore, multiple vaccine constructs have shown promise, including two that have an approximately 95% protective efficacy against COVID-19. However, these interventions were directed against the initial SARS-CoV-2 virus that emerged in 2019. The recent detection of SARS-CoV-2 variants B.1.1.7 in the UK and B.1.351 in South Africa is of concern because of their purported ease of transmission and extensive mutations in the spike protein. Here we show that B.1.1.7 is refractory to neutralization by most monoclonal antibodies against the N-terminal domain of the spike protein and is relatively resistant to a few monoclonal antibodies against the receptor-binding domain. It is not more resistant to plasma from individuals who have recovered from COVID-19 or sera from individuals who have been vaccinated against SARS-CoV-2. The B.1.351 variant is not only refractory to neutralization by most monoclonal antibodies against the N-terminal domain but also by multiple individual monoclonal antibodies against the receptor-binding motif of the receptor-binding domain, which is mostly due to a mutation causing an E484K substitution. Moreover, compared to wild-type SARS-CoV-2, B.1.351 is markedly more resistant to neutralization by convalescent plasma (9.4-fold) and sera from individuals who have been vaccinated (10.3-12.4-fold). B.1.351 and emergent variants with similar mutations in the spike protein present new challenges for monoclonal antibody therapies and threaten the protective efficacy of current vaccines.