In the past few years, major advances have been achieved in understanding the nature and the maintenance mechanisms of the HIV reservoir. Although antiretroviral therapy works well in a majority of patients, it faces problems of compliance, resistance, toxicity, and cost. In most cases, the remaining HIV reservoir precluding antiretroviral cessation consists of a tiny cell pool that is long-lived and inaccessible to current therapies. New strategies are therefore needed to either purge or control this residual reservoir and finally stop antiretroviral drugs. Both ways leading to a functional or a sterilizing cure are currently pursued. Several molecules have been identified to achieve these goals and some of them have already entered clinical testing in humans. In this article, we review recent findings on the biology of HIV persistence and detail how HIV eradication trials should be designed in the near future.