HIV’s ability to establish latent reservoirs of reactivation-competent virus is the major barrier to cure. “Shock and kill” methods consisting of latency-reversing agents (LRAs) followed by elimination of reactivating cells through cytopathic effects are under active development. However, the clinical efficacy of LRAs remains to be established. Moreover, recent studies indicate that reservoirs may not be reduced efficiently by either viral cytopathic or CD8⁺ T-cell-mediated mechanisms. In this perspective, we highlight challenges to T-cell-mediated elimination of HIV reservoirs, including characteristics of responding T cells, aspects of the cellular reservoirs, and properties of the latent virus itself. We also discuss potential strategies to overcome these challenges by targeting the antiviral activity of T cells toward appropriate viral antigens following latency.