Antiretroviral therapy (ART) has markedly improved the life-expectancy of people living with HIV. However, during both HIV infection of humans and simian immunodeficiency virus infection of macaques, virus replication almost invariably rebounds upon ART interruption, due to the long-term persistency of a pool of latently infected cells harbouring integrated, replication-competent virus (known as the virus reservoir). Solving this 'HIV reservoir problem' is the key to achieving a cure (or at least a persistent remission) for HIV infection. Here, we summarize the key scientific evidence supporting the hypothesis that host immune responses, including those mediated by CD8