Since the start of the coronavirus disease-2019 (COVID-19) pandemic, severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) has caused more than 2 million deaths worldwide. Multiple vaccines have been deployed to date, but the continual evolution of the viral receptor-binding domain (RBD) has recently challenged their efficacy. In particular, SARS-CoV-2 variants originating in the U.K. (B.1.1.7), South Africa (B.1.351) and New York (B.1.526) have reduced neutralization activity from convalescent sera and compromised the efficacy of antibody cocktails that received emergency use authorization. Whereas vaccines can be updated periodically to account for emerging variants, complementary strategies are urgently needed to avert viral escape. One potential alternative is the use of camelid VHHs (also known as nanobodies), which due to their small size can recognize protein crevices that are inaccessible to conventional antibodies. Here, we isolate anti-RBD nanobodies from llamas and "nanomice" we engineered to produce VHHs cloned from alpacas, dromedaries and camels. Through binding assays and cryo-electron microscopy, we identified two sets of highly neutralizing nanobodies. The first group expresses VHHs that circumvent RBD antigenic drift by recognizing a region outside the ACE2-binding site that is conserved in coronaviruses but is not typically targeted by monoclonal antibodies. The second group is almost exclusively focused to the RBD-ACE2 interface and fails to neutralize pseudoviruses carrying the E484K or N501Y substitutions. Notably however, they do neutralize the RBD variants when expressed as homotrimers, rivaling the most potent antibodies produced to date against SARS-CoV-2. These findings demonstrate that multivalent nanobodies overcome SARS-CoV-2 variant 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.