The persistence of latent, replication-competent HIV-1 proviruses in resting CD4+ T cells, and other cellular reservoirs, represents a major barrier to a cure. This reservoir is impervious to the immune system and to antiretroviral therapy, but has the potential to produce infectious rebound virus if antiretroviral therapy is interrupted. There are multiple ongoing efforts to identify and/or develop novel therapeutic strategies to eliminate or silence this latent reservoir of HIV-1 infection. One of these strategies is termed “block and lock”. The “block” refers to a therapeutic agent’s capacity to inhibit (or “block”) transcription of HIV-1 proviruses, while the “lock” refers to its capacity to induce permanent silencing of the proviruses, typically via repressive epigenetic modifications. The “block and lock” approach elicits a functional, rather than sterilizing, cure for HIV-1 infection. This review article focuses on therapeutic approaches (i.e., small molecules, nucleic acids and recombinant proteins) that have been identified to block and, in some cases, lock HIV-1 in the latent state. We also touch on critical research that needs to be accomplished to advance this approach into humans.