Recent studies demonstrate that suppressive therapy can drive HIV-1 RNA levels to less than 50 copies mL−1 in patient plasma. Yet, ultrasensitive assays show that most patients continue to harbour low-level persistent viremia. Treatment intensification studies indicate that low-level viremia could arise from several different sources. These sources include: (i) long-lived HIV-infected cells that replicate and produce virus; (ii) ongoing replication cycles in cells located in sanctuary sites where drug levels are suboptimal; and/or (iii) proliferation of latently infected cells with regeneration of a stable reservoir of slowly dividing infected cells. A well-defined latent reservoir of HIV is memory CD4+ T-cells where latency is established when an activated CD4+ T-cell becomes infected by HIV, but transitions to a terminally differentiated memory cell before it is eliminated. This review examines the dynamics and possible reservoirs of persistent HIV in patients on suppressive therapy, the mechanisms promoting viral latency and strategies to purge latent viral reservoirs. The promising research described here takes a number of steps forward to seriously address HIV remission and/or eradication.