The study of phase transitions is one of the fundamental problems of physics. The goal of this seminar was to understand better the spectacular progress made recently in constructing continuum models. Concentrating on a few examples such as the microstructure of crystals, defects in liquid crystals and liquid-vapor interfaces, several key points are described, for example the structure and evolution of the interfaces, regularization via interfacial energy, and equilibrium theories. The mathematical treatment of these questions involves large-oscillation theories (Young’s measures, compensated compactness), spectral theory, admissibility of shock waves, long-time behavior of dynamical systems, high-order perturbations, group actions, solitons, and others.