One of the main challenges in the fight against HIV infection is to develop strategies that are able to eliminate the persistent viral reservoir that harbours integrated, replication-competent provirus within host cellular DNA. This reservoir is resistant to antiretroviral therapy (ART) and to clearance by the immune system of the host; viruses originating from this reservoir lead to rebound viraemia once treatment is stopped, giving rise to new rounds of infection. Several studies have focused on elucidating the cells and tissues that harbour persistent virus, the true size of the reservoir and how best to target it, but these topics are the subject of ongoing debate. In this Viewpoint article, several experts in the field discuss the constitution of the viral reservoir, how best to measure it and the best ways to target this source of persistent infection.