T-cell immunity is important for recovery from COVID-19 and provides heightened immunity for re-infection. However, little is known about the SARS-CoV-2-specific T-cell immunity in virus-exposed individuals. Here we report virus-specific CD4+ and CD8+ T-cell memory in recovered COVID-19 patients and close contacts. We also demonstrate the size and quality of the memory T-cell pool of COVID-19 patients are larger and better than those of close contacts. However, the proliferation capacity, size and quality of T-cell responses in close contacts are readily distinguishable from healthy donors, suggesting close contacts are able to gain T-cell immunity against SARS-CoV-2 despite lacking a detectable infection. Additionally, asymptomatic and symptomatic COVID-19 patients contain similar levels of SARS-CoV-2-specific T-cell memory. Overall, this study demonstrates the versatility and potential of memory T cells from COVID-19 patients and close contacts, which may be important for host protection.