Phase Transitions and Hysteresis : Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme Italy July 13–21 1993 1) Phase Transitions represented by generalizations of the classical Stefan problem. This is studied by Kenmochi and Rodrigues by means of variational techniques. 2) Hysteresis Phenomena. Some alloys exhibit shape memory effects corresponding to a stress strain relation which strongly depends on temperature; mathematical physical aspects are treated in Müllers paper. In a general framework hysteresis can be described by means of hysteresis operators in Banach spaces of time dependent functions; their properties are studied by Brokate. 3) Numerical analysis. Several models of the phenomena above can be formulated in terms of nonlinear parabolic equations. Here Verdi deals with the most updated approximation techniques