Non-human primate models will expedite therapeutics and vaccines for coronavirus disease 2019 (COVID-19) to clinical trials. Here, we compare acute severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection in young and old rhesus macaques, baboons and old marmosets. Macaques had clinical signs of viral infection, mild to moderate pneumonitis and extra-pulmonary pathologies, and both age groups recovered in two weeks. Baboons had prolonged viral RNA shedding and substantially more lung inflammation compared with macaques. Inflammation in bronchoalveolar lavage was increased in old versus young baboons. Using techniques including computed tomography imaging, immunophenotyping, and alveolar/peripheral cytokine response and immunohistochemical analyses, we delineated cellular immune responses to SARS-CoV-2 infection in macaque and baboon lungs, including innate and adaptive immune cells and a prominent type-I interferon response. Macaques developed T-cell memory phenotypes/responses and bystander cytokine production. Old macaques had lower titres of SARS-CoV-2-specific IgG antibody levels compared with young macaques. Acute respiratory distress in macaques and baboons recapitulates the progression of COVID-19 in humans, making them suitable as models to test vaccines and therapies.