Thirty-five years after the identification of HIV-1 as the causative agent of AIDS, we are still in search of vaccines and treatments to eradicate this devastating infectious disease. Progress has been made in understanding the molecular pathogenesis of this infection, which has been crucial for the development of the current therapy regimens. However, despite their efficacy at limiting active viral replication, these drugs are unable to purge the latent reservoir: a pool of cells that harbor transcriptionally inactive, but replication-competent HIV-1 proviruses, and that represent the main barrier to eradicate HIV-1 from affected individuals. In this review, we discuss advances in the field that have allowed a better understanding of HIV-1 latency, including the diverse cell types that constitute the latent reservoir, factors influencing latency, tools to study HIV-1 latency, as well as current and prospective therapeutic approaches to target these latently infected cells, so a functional cure for HIV/AIDS can become a reality.