This research school is about geometric methods in dynamics. Modern dynamical systems and geometry benefit from the close interactions between both theories; in particular geometry helps to understand global aspects of dynamical systems. Our main goal is to bring out some of these connections in elementary and beautiful examples such as: the precise count of geodesics on surfaces of negative curvature, the count of periodic orbits of Hamiltonian systems on tori, and the existence of periodic orbits of a prescribed energy of classical mechanical systems.