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People living with HIV (PLWH) can exhibit impaired immune responses to vaccines. Accumulating evidence indicates that PLWH, particularly those receiving antiretroviral therapy, mount strong antibody responses to COVID-19 vaccines, but fewer studies have examined cellular immune responses to the vaccinations. Here, we used an activation-induced marker (AIM) assay to quantify SARS-CoV-2 spike-specific CD4+ and CD8+ T cells generated by two and three doses of COVID-19 vaccines in 50 PLWH receiving antiretroviral therapy, compared to 87 control participants without HIV. In a subset of PLWH, T-cell responses were also assessed after post-vaccine breakthrough infections and/or receipt of a fourth vaccine dose. All participants remained SARS-CoV-2 infection-naive until at least one month after their third vaccine dose. SARS-CoV-2 infection was determined by seroconversion to a Nucleocapsid (N) antigen, which occurred in 21 PLWH and 38 control participants after the third vaccine dose. Multivariable regression analyses were used to investigate the relationships between sociodemographic, health- and vaccine-related variables, vaccine-induced T-cell responses, and breakthrough infection risk. We observed that a third vaccine dose boosted spike-specific CD4+ and CD8+ T-cell frequencies significantly above those measured after the second dose (all p < 0.0001). Median T-cell frequencies did not differ between PLWH and controls after the second dose ( p > 0.1), but CD8+ T-cell responses were modestly lower in PLWH after the third dose ( p = 0.02), an observation that remained significant after adjusting for sociodemographic, health- and vaccine-related variables ( p = 0.045). In PLWH who experienced a breakthrough infection, median T-cell frequencies increased even higher than those observed after three vaccine doses ( p < 0.03), and CD8+ T-cell responses in this group remained higher even after a fourth vaccine dose ( p = 0.03). In multivariable analyses, the only factor associated with an increased breakthrough infection risk was younger age, which is consistent with the rapid increase in SARS-CoV-2 seropositivity that was seen among younger adults in Canada after the initial appearance of the Omicron variant. These results indicate that PLWH receiving antiretroviral therapy mount strong T-cell responses to COVID-19 vaccines that can be enhanced by booster doses or breakthrough infection.