Combination antiretroviral therapy (cART) has led to a reduction in morbidity and mortality in HIV-infected patients but therapy is lifelong and there is no cure for HIV. The major barriers to cure include HIV latency, which has been identified in different T-cell subsets, as well as persistence of HIV in anatomical reservoirs. We review recent developments in our understanding of the major reservoirs of HIV in patients on cART as well as how latency is established and maintained in T cells. Finally, we review the scientific rationale of and clinical experience with pharmacotherapeutic strategies aimed at eliminating latently infected cells.