Interventions against variants of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) are urgently needed. Stable and potent nanobodies (Nbs) that target the receptor binding domain (RBD) of SARS-CoV-2 spike are promising therapeutics. However, it is unknown if Nbs broadly neutralize circulating variants. We found that RBD Nbs are highly resistant to variants of concern (VOCs). High-resolution cryoelectron microscopy determination of eight Nb-bound structures reveals multiple potent neutralizing epitopes clustered into three classes: Class I targets ACE2-binding sites and disrupts host receptor binding. Class II binds highly conserved epitopes and retains activity against VOCs and RBD_SARS-CoV. Cass III recognizes unique epitopes that are likely inaccessible to antibodies. Systematic comparisons of neutralizing antibodies and Nbs provided insights into how Nbs target the spike to achieve high-affinity and broadly neutralizing activity. Structure-function analysis of Nbs indicates a variety of antiviral mechanisms. Our study may guide the rational design of pan-coronavirus vaccines and therapeutics.