Self versus non-self discrimination is at the core of T-lymphocyte recognition. To this end, αβ T-cell receptors (TCRs) ligate ‘foreign’ peptides bound to major histocompatibility complex (MHC) class I or class II molecules (pMHC) arrayed on the surface of antigen-presenting cells (APCs). Since the discovery of TCRs approximately 30 years ago, considerable structural and functional data have detailed the molecular basis of their extraordinary ligand specificity and sensitivity in mediating adaptive T-cell immunity. This review focuses on the structural biology of the Fab-like TCRαβ clonotypic heterodimer and its unique features in conjunction with those of the associated CD3εγ and CD3εδ heterodimeric molecules, which, along with CD3ζζ homodimer, comprise the TCR complex in a stoichiometry of 1:1:1:1. The basis of optimized TCRαβ docking geometry on the pMHC linked to TCR mechanotransduction and required for T-cell signaling as well as CD4 and CD8 co-receptor function is detailed. A model of the TCR ectodomain complex including its connecting peptides suggests how force generated during T-cell immune surveillance and at the immunological synapse results in dynamic TCR quaternary change involving its heterodimeric components. Potential insights from the structural biology relevant to immunity and immunosuppression are revealed.