In the global fight against HIV/AIDS, the persistence of viral reservoirs, established early during primary infection by the Human Immunodeficiency virus (HIV), is probably the biggest challenge to finding a cure for HIV, despite the major progress provided by antiviral treatments. This persistence reflects virus latency in the immune system and imposes the prospect of life-long treatment with risk of toxicities, and major costs for society. Innovative strategies aimed at purging these viral reservoirs are therefore required, but the complexity of the latency mechanisms has made a sterilization cure difficult to realize until now. However, recent cases of HIV remission have encouraged the research community, and a functional cure seems to be a more realistic goal to attempt. A better understanding of the mechanisms responsible for such latency has been provided by basic research, although recent attempts at purging the reservoirs by disrupting latency have been somewhat disappointing. In contrast, the early initiation of anti-retroviral treatment appears to be extremely beneficial for HIV remission, as recently suggested by the Mississippi baby case or by the series of post-treatment controllers, respectively. Early treatment initiation can indeed rapidly limit the size of viral reservoirs, or hamper their formation and preserve the host’s immune responses. Despite these encouraging results, it is nevertheless important to deploy more effort toward a better understanding of latency mechanisms and to the development of innovative approaches and molecules in the hope of achieving a cure for HIV.