International audienceHIV research has made rapid progress and led to remarkable achievements in recent decades, the most important of which are combination antiretroviral therapies (cART). However, in the absence of a vaccine, the pandemic continues, and additional strategies are needed. The 'towards an HIV cure' initiative aims to eradicate HIV or at least bring about a lasting remission of infection during which the host can control viral replication in the absence of cART. Cases of spontaneous and treatment-induced control of infection offer substantial hope. Here, we describe the scientific knowledge that is lacking, and the priorities that have been established for research into a cure. We discuss in detail the immunological lessons that can be learned by studying natural human and animal models of protection and spontaneous control of viraemia or of disease progression. In particular, we describe the insights we have gained into the immune mechanisms of virus control, the impact of early virus-host interactions and why chronic inflammation, a hallmark of HIV infection, is an obstacle to a cure. Finally, we enumerate current interventions aimed towards improving the host immune response