Since the start of the COVID-19 pandemic, SARS-CoV-2 has caused millions of deaths worldwide. While many vaccines have been deployed to date, the continual evolution of the viral receptor-binding domain (RBD) has challenged their efficacy. In particular, emerging variants B.1.1.7 (U.K.), B.1.351 (South Africa) and P.1 (Brazil) have compromised convalescent sera and immunotherapies that received emergency use authorization1–3. One potential alternative to avert viral escape is the use of camelid VHHs or nanobodies, which can recognize epitopes often inaccessible to conventional antibodies4. Here, we isolate anti-RBD nanobodies from llamas and “nanomice” we engineered to produce VHHs cloned from alpacas, dromedaries and camels. We identified two sets of highly neutralizing nanobodies. Group 1 circumvents antigenic drift by recognizing an RBD region that is highly conserved in coronaviruses but rarely targeted by human antibodies. Group 2 is almost exclusively focused to the RBD-ACE2 interface and fails to neutralize variants carrying E484K or N501Y substitutions. Notably however, group 2 nanobodies retain full neutralization activity against variants when expressed as homotrimers, rivaling the most potent antibodies produced to date against SARS-CoV-2. These findings suggest that multivalent nanobodies overcome SARS-CoV-2 mutations through two separate mechanisms: enhanced avidity for the ACE2 binding domain, and recognition of conserved epitopes largely inaccessible to human antibodies. Therefore, while new SARS-CoV-2 mutants will continue to emerge, nanobodies represent promising tools to prevent COVID-19 mortality when vaccines are compromised.