The recently emerged SARS-CoV-2 Omicron variant encodes 37 amino acid substitutions in the spike protein, 15 of which are in the receptor-binding domain (RBD), thereby raising concerns about the effectiveness of available vaccines and antibody-based therapeutics. Here we show that the Omicron RBD binds to human ACE2 with enhanced affinity, relative to the Wuhan-Hu-1 RBD, and binds to mouse ACE2. Marked reductions in neutralizing activity were observed against Omicron compared to the ancestral pseudovirus in plasma from convalescent individuals and from individuals who had been vaccinated against SARS-CoV-2, but this loss was less pronounced after a third dose of vaccine. Most monoclonal antibodies that are directed against the receptor-binding motif lost in vitro neutralizing activity against Omicron, with only 3 out of 29 monoclonal antibodies retaining unaltered potency, including the ACE2-mimicking S2K146 antibody. Furthermore, a fraction of broadly neutralizing sarbecovirus monoclonal antibodies neutralized Omicron through recognition of antigenic sites outside the receptor-binding motif, including sotrovimab, S2X259 and S2H97. The magnitude of Omicron-mediated immune evasion marks a major antigenic shift in SARS-CoV-2. Broadly neutralizing monoclonal antibodies that recognize RBD epitopes that are conserved among SARS-CoV-2 variants and other sarbecoviruses may prove key to controlling the ongoing pandemic and future zoonotic spillovers.