T-cells are a vital type of white blood cell that circulate around our bodies, scanning for cellular abnormalities and infections. They recognise disease-associated antigens via a surface receptor called the T-cell antigen receptor (TCR). If there were a specific TCR for every single antigen, no mammal could possibly contain all the T-cells it needs. This is clearly absurd and suggests that T-cell recognition must, to the contrary, be highly degenerate. Yet highly promiscuous TCRs would appear to be equally impossible: they are bound to recognise self as well as non-self antigens. We review how contributions from mathematical analysis have helped to resolve the paradox of the promiscuous TCR. Combined experimental and theoretical work shows that TCR degeneracy is essentially dynamical in nature, and that the T-cell can differentially adjust its functional sensitivity to the salient epitope, "tuning up" sensitivity to the antigen associated with disease and "tuning down" sensitivity to antigens associated with healthy conditions. This paradigm of continual modulation affords the TCR repertoire, despite its limited numerical diversity, the flexibility to respond to almost any antigenic challenge while avoiding autoimmunity.Keywords: T-cell activation, T-cell antigen receptor, mathematical modelling, TCR repertoire, co-receptor, costimulation 1 {biographical notes} Hugo van den Berg developed mathematical models of energy budgets at the Free University of Amsterdam, where he obtained a PhD in 1998. His current research interests include endocrinology and energetics, mathematical models of the immune system, and the development of the excitability of the uterine muscle wall during gestation.