The World Health Organization has called for increased molecular testing in response to the COVID-19 pandemic, but different countries have taken very different approaches. We used a simple mathematical model to investigate the potential effectiveness of alternative testing strategies for COVID-19 control. Weekly screening of healthcare workers (HCWs) and other at-risk groups using PCR or point-of-care tests for infection irrespective of symptoms is estimated to reduce their contribution to transmission by 25-33%, on top of reductions achieved by self-isolation following symptoms. Widespread PCR testing in the general population is unlikely to limit transmission more than contact-tracing and quarantine based on symptoms alone, but could allow earlier release of contacts from quarantine. Immunity passports based on tests for antibody or infection could support return to work but face significant technical, legal and ethical challenges. Testing is essential for pandemic surveillance but its direct contribution to the prevention of transmission is likely to be limited to patients, HCWs and other high-risk groups.