Despite antiretroviral therapy (ART),humanimmunodeficiency virus (HIV)-1 persists in a stable latent reservoir1,2, primarily in resting memory CD41 T cells3,4. This reservoir presents a major barrier to the cure of HIV-1 infection. To purge the reservoir, pharmacological reactivation of latent HIV-1 has been proposed5 and tested both in vitro and in vivo6-8. A key remaining question is whether virusspecific immune mechanisms, including cytotoxic T lymphocytes (CTLs), can clear infected cells in ART-treated patients after latency is reversed. Here we show that there is a striking all or none pattern for CTL escape mutations in HIV-1 Gag epitopes. Unless ART is started early, the vast majority (.98%) of latent viruses carry CTL escapemutations that render infectedcells insensitive toCTLsdirected at common epitopes. To solve this problem, we identified CTLs that could recognize epitopes fromlatent HIV-1 that were unmutated in every chronically infected patient tested. Upon stimulation, these CTLs eliminated target cells infected with autologous virus derived fromthe latent reservoir, bothinvitro andinpatient-derived humanized mice. The predominance of CTL-resistant viruses in the latent reservoir poses a major challenge to viral eradication. Our results demonstrate that chronically infected patients retain a broad-spectrum viral-specific CTL response and that appropriate boosting of this response may be required for the elimination of the latent reservoir.