The adaptive immune system is composed of specialised cells that detect and respond to foreign pathogens. T cells and B cells, the main coordinators of this immune response, present lymphocyte antigen receptors on their cell surface that can bind to antigens of foreign pathogens. Lymphocyte antigen receptors encompasses the T-cell receptor (TCR) on the surface of T cells, which bind peptide antigens presented by MHC complexes (peptide-MHC) on the surface of cells. The second type of antigen receptors are immunoglobulins, which either occur as surface expressed B-cell receptor (BCR) or secreted antibodies by B cells. This thesis investigates how the biophysical properties of an interaction between antigen receptors and their antigen is translated into a functional cellular response. Specifically, the focus is on how the TCR discriminates between foreign and endogenous (self) antigens. In a second project the mechanism of antibody/antigen binding and its impact of antibody function is examined. This involved developing advanced SPR methods to measure ultra-low affinity interaction and bivalent binding. The affinity measurements are then compared with the functional responses the antigens evoke by measuring T-cell activation and the neutralisation potency of antibodies. Project 1: Antigen Discrimination by the TCR: T cells use their TCRs to discriminate between lower-affinity self and higher-affinity foreign antigens. Early studies on murine TCRs showed that TCRs are capable of perfectly discriminating between small difference in antigens. This means that antigens with a slight change in structure and binding affinity compared to the cognate, stimulatory antigen were unable to activate T cells, regardless of their concentration. The concept of perfect antigen discrimination cannot be reconciled with recent clinical evidence on T-cell off-target toxicity, and autoimmune responses. To understand this apparent discrepancy, this thesis revisits antigen discrimination by the TCR. The discrimination strength is quantified from previously published data and by remeasuring affinities of the OT-I-TCR to antigen variants. This analysis reveals that the discrimination strength is significantly lower than the current consensus, which means low-affinity antigens can induce an immune response when present at high concentrations. Though lower, TCRs still have enhanced discrimination compared to other surface ligand receptors. This thesis also reports the first affinity measurements of TCR binding naturally occurring self-peptide-MHCs at physiological temperatures. Project 2: Bivalent Antibody Binding: Antibodies carry two identical binding sites, which allow them to achieve strong binding to their antigen through bivalent binding and thus increase their neutralisation potency. Although antibodies bind bivalently, current methods to screen and optimise them rely on methods that measure monovalent binding. However monovalent binding parameters have failed to predict the neutralisation potency of antibodies. Interestingly, methods to measure bivalent binding using SPR are readily available but there are currently no mathematical models that can be used to analyse this bivalent data. This thesis presents a particle-based bivalent model that relies on stochastic-spatial simulations to analyse bivalent binding data. The model includes a new biophysical parameter, termed the "molecular reach", that quantifies the maximum separation distance between two antigens that still allows a single antibody to bind them both simultaneously. The model is used to determine the binding kinetics of a panel of 80 antibodies isolated from COVID-19 patients that recognise the receptor-binding-domain (RBD) of SARS-CoV-2. This revealed that the molecular reach of an antibody is the strongest correlate of SARS-CoV-2 viral neutralisation. Using the bivalent binding parameters, the antibody concentrations required for viral neutralisation were predicted.