The adaptive immune system recognizes pathogen- and cancer-specific features and is endowed with memory, enabling it to respond quickly and efficiently to repeated encounters with the same antigens. T cells play a central role in the adaptive immune system by directly targeting intracellular pathogens and helping to activate B cells to secrete antibodies. Several fundamental protein interactions-including those between major histocompatibility complex (MHC) proteins and antigen-derived peptides as well as between T cell receptors and peptide-MHC complexes-underlie the ability of T cells to recognize antigens with great precision. Computational approaches to predict these interactions are increasingly being used for medically relevant applications, including vaccine design and prediction of patient response to cancer immunotherapies. We provide computational researchers with an accessible introduction to the adaptive immune system, review computational approaches to predict the key protein interactions underlying T cell-mediated adaptive immunity, and highlight remaining challenges.