This volume contains the lecture notes written by the four principal speakers at the CIME session on Dynamical Systems held at Montecatini, Italy in June 1994. The goal of the session was to illustrate how methods of dynamical systems can be applied to the study of ordinary and partial differential equations. Topics in random differential equations, singular perturbations, the Conley index theory and non-linear PDEs were discussed. Readers interested in asymptotic behaviour of solutions of ODEs and PDEs who are familiar with basic notions of dynamical systems should find this text useful.