T cells are involved in the early identification and clearance of viral infections and also support the development of antibodies by B cells. This central role for T cells makes them a desirable target for assessing the immune response to SARS-CoV-2 infection. Here, we combined two high-throughput immune profiling methods to create a quantitative picture of the T-cell response to SARS-CoV-2. First, at the individual level, we deeply characterized 3 acutely infected and 58 recovered COVID-19 subjects by experimentally mapping their CD8 T-cell response through antigen stimulation to 545 Human Leukocyte Antigen (HLA) class I presented viral peptides (class II data in a forthcoming study). Then, at the population level, we performed T-cell repertoire sequencing on 1,815 samples (from 1,521 COVID-19 subjects) as well as 3,500 controls to identify shared "public" T-cell receptors (TCRs) associated with SARS-CoV-2 infection from both CD8 and CD4 T cells. Collectively, our data reveal that CD8 T-cell responses are often driven by a few immunodominant, HLA-restricted epitopes. As expected, the T-cell response to SARS-CoV-2 peaks about one to two weeks after infection and is detectable for at least several months after recovery. As an application of these data, we trained a classifier to diagnose SARS-CoV-2 infection based solely on TCR sequencing from blood samples, and observed, at 99.8% specificity, high early sensitivity soon after diagnosis (Day 3-7 = 85.1% [95% CI = 79.9-89.7]; Day 8-14 = 94.8% [90.7-98.4]) as well as lasting sensitivity after recovery (Day 29+/convalescent = 95.4% [92.1-98.3]). These results demonstrate an approach to reliably assess the adaptive immune response both soon after viral antigenic exposure (before antibodies are typically detectable) as well as at later time points. This blood-based molecular approach to characterizing the cellular immune response has applications in clinical diagnostics as well as in vaccine development and monitoring.