Book excerpt: HIV-1 eradication from infected individuals has not been achieved with the use of highly active antiretroviral therapy (HAART) for a prolonged period of time. The cellular reservoir for HIV-1 in resting memory CD4 T cells remains a major obstacle to viral elimination. The reservoir does not decay significantly over long periods of time as is able to release replication competent HIV-1 upon cell activation. Residual ongoing viral replication may likely occur in many patients because low levels of virus can be detected in plasma by sensitive assays and transient episodes of viremia, or HIV-1 blips, are often observed in patients even with successful viral suppression for many years. Here we review our current knowledge of the factors contributing to viral persistence, the latent reservoir, and blips, and mathematical models developed to explore them and their relationships. We show how mathematical modeling can help improve our understanding of HIV-1 dynamics in patients on HAART and the quantitative events underlying HIV-1 latency, reservoir stability, low-level viremic persistence, and emergence of intermittent viral blips. We also discuss treatment implications related to these studies.