Combination antiretroviral therapy (ART) for HIV-1 infection reduces plasma virus levels to below the limit of detection of clinical assays. However, even with prolonged suppression of viral replication with ART, viremia rebounds rapidly after treatment interruption. Thus, ART is not curative. The principal barrier to cure is a remarkably stable reservoir of latent HIV-1 in resting memory CD4+ T cells. In this review, we consider explanations for the remarkable stability of the latent reservoir. Stability does not appear to reflect replenishment from new infection events but rather normal physiologic processes that provide for immunologic memory. Of particular importance are proliferative processes that drive clonal expansion of infected cells. Recent evidence suggests that in some infected cells, proliferation is a consequence of proviral integration into host genes associated with cell growth. Efforts to cure HIV-1 infection by targeting the latent reservoir may need to consider the potential of latently infected cells to proliferate.