We review aspects of the antibody response to SARS-CoV-2, the causative agent of the COVID-19 pandemic. The topics we cover are relevant to immunotherapy with plasma from recovered patients, monoclonal antibodies against the viral S-protein, and soluble forms of the receptor for the virus, angiotensin converting enzyme 2. The development of vaccines against SARS-CoV-2, an essential public health tool, will also be informed by an understanding of the antibody response in infected patients. Although virus-neutralizing antibodies are likely to protect, antibodies could potentially trigger immunopathogenic events in SARS-CoV-2-infected patients or enhance infection. An awareness of these possibilities may benefit clinicians and the developers of antibody-based therapies and vaccines.