The recognition of foreign antigen by T cells is the foundation of the adaptive immune response, and this has been demonstrated experimentally to be an extremely sensitive, specific and reliable process. In this thesis, models for T cell signalling based on recent experimental data are constructed to understand how these properties arise. The thesis first reviews the biological and mathematical background necessary to model T cell activation. Next, a stochastic interpretation of the standard model for TCR specificity (McKeithan's kinetic proofreading model) is made, which extends the analysis from TCR concentrations to individual TCR. When stochastic ligand dissociation is included, the analysis shows that kinetic proofreading alone fails to provide the necessary degree of specificity seen experimentally. Based on this analysis, a new model that incorporates the essential elements of proofreading (i.e., delay followed by activation) and is more consistent with known TCR signalling biology is constructed. This new model predicts a role for the immune synapse and self ligands in amplifying and sustaining T cell signalling, as well as a novel role for multiple ITAMs to decrease the variance of the activation threshold. The next model moves from the level of individual TCR to study interactions between a population of receptors. A Monte Carlo simulation of a lattice of TCR interacting with ligands is constructed, which integrates the most important models for T cell specificity (kinetic proofreading) and sensitivity (serial ligation), and incorporates recent evidence for cross-talk between neighbouring receptors. This simulation reveals that the specificity of T cell ligand discrimination can be significantly enhanced with receptor cooperativity. Finally, the model suggests a resolution to the paradox of positive and negative selection on a similar set of ligands, and uses this to explain the surprising repertoire of transgenic mice that express the same peptide on all MHC II molecules.