Monoclonal anti-SARS-CoV-2 immunoglobulins represent a treatment option for COVID-19. However, their production in mammalian cells is not scalable to meet the global demand. Single-domain (VHH) antibodies (also called nanobodies) provide an alternative suitable for microbial production. Using alpaca immune libraries against the receptor-binding domain (RBD) of the SARS-CoV-2 Spike protein, we isolated 45 infection-blocking VHH antibodies. These include nanobodies that can withstand 95°C. The most effective VHH antibody neutralizes SARS-CoV-2 at 17-50 pM concentration (0.2-0.7 µg per liter), binds the open and closed states of the Spike, and shows a tight RBD interaction in the X-ray and cryo-EM structures. The best VHH trimers neutralize even at 40 ng per liter. We constructed nanobody tandems and identified nanobody monomers that tolerate the K417N/T, E484K, N501Y, and L452R immune-escape mutations found in the Alpha, Beta, Gamma, Epsilon, Iota, and Delta/Kappa lineages. We also demonstrate neutralization of the Beta strain at low-picomolar VHH concentrations. We further discovered VHH antibodies that enforce native folding of the RBD in the E. coli cytosol, where its folding normally fails. Such "fold-promoting" nanobodies may allow for simplified production of vaccines and their adaptation to viral escape-mutations.