Antiretroviral therapy (ART) has brought the HIV/AIDS epidemic under control, but a curative strategy for viral eradication is still needed. The cessation of ART results in rapid viral rebound from latently infected CD4+ T cells, showing that control of viral replication alone does not fully restore immune function, nor does it eradicate viral reservoirs. With a better understanding of factors and mechanisms that promote viral latency, current approaches are primarily focused on the permanent silencing of latently infected cells (“block and lock”) or reactivating HIV-1 gene expression in latently infected cells, in combination with immune restoration strategies to eliminate HIV infected cells from the host (“shock and kill”). In this review, we provide a summary of the current, most promising approaches for HIV-1 cure strategies, including an analysis of both latency-promoting agents (LPA) and latency-reversing agents (LRA) that have shown promise in vitro, ex vivo, and in human clinical trials to reduce the HIV-1 reservoir.