Brake squeal remains a topic of ongoing research despite much progress being made especially in the last decade. Many of the feasible brake squeal mechanisms are far from being fully understood and most numerical methods for predicting brake squeal propensity are not reliable or practical to be used in industry. In this thesis, a range of methods, from nonlinear time series analysis to structural finite element and acoustic boundary element simulations, are applied to the analysis of brake squeal with a focus on treating it as a nonlinear problem. The relationship between the time-averaged friction coefficient (μt) and the peak sound pressure level (SPLp) obtained for a full brake system in a noise dynamometer is analysed by statistical and correlation techniques. Results show that the noise performance of different pad configurations can be ranked by the degree of nonlinearity in the correlation between μt and SPLp. The role of nonlinearity in brake squeal is further examined by applying some tools of nonlinear dynamics to the squeal signals. Several dynamic regimes have been identified in the squeal signals: transiting from a limit cycle to an unstable torus attractor and the Ruelle-Takens-Newhouse route to chaos is revealed. Simplified brake systems in the form of pad-on-plate and pad-on-disc models are considered. Structural finite element/acoustic boundary element simulations are used to study the development of vibration instabilities and their acoustic radiation. As brake squeal is a transient phenomenon, changes in the operating conditions are simulated for a range of friction coefficients and pressures, temperature-dependent and pressuredependent lining material properties. Although in-plane pad modes are not predicted to be unstable by the conventional finite element complex eigenvalue analysis, they are shown to initiate instability and couple with the out-of-plane plate/disc motion to cause potential squeal away from the resonance frequency of the plate/disc. The frequencies of pad modes have been found to vary significantly, depending on the pressure and the friction coefficient. The kinetic energy of pad modes increases with both pressure and friction coefficient and the dissipated energy is negative (i.e., energy source). Pad modes have been found to radiate high acoustic power level and to influence the acoustic radiation of adjacent disc modes. The pad-on-plate model has been successful in simulating an experiment in which the so-called instantaneous mode squeal is observed. By using nonlinear time series analysis, intermittent bursts of high-amplitude, out-of-plane pad vibrations have been shown to drive the disc s out-of-plane dynamics into toroidal chaos. This result supports the two squeal mechanisms identified in this study: (i) nonlinearity and the route to chaos; and (ii) instabilities initiated by in-plane pad modes (due to changes in operating conditions and material properties).