The paper covers the development and application of finite element techniques to thermo-mechanical instability in braking and brake disc hot judder. After a brief review of the phenomena of thermo-mechanical instability in braking and disc thermal judder, this paper describes and analyses experimental results of hot judder testing carried out on a dynamometer. Development of axisymmetric and 3D thermo-mechanical FE models capable of dealing with localised heat generation and wear is then described, with examples to illustrate their application to hot judder simulation. Through use of the 3D thermo-mechanical simulation together with the detailed analysis of experimental results, it is found that the out-of-phase thermoelastic "buckling instability" is a significant factor causing hot spotting and hot judder. This new finding contributes to the fundamental understanding of the phenomena and widens the scope to improve the brake system performance in the relevant aspects.