The predominance of linear models in systems theory has tended to obscure the natural structure possessed by given nonlinear physical systems, either through linearisation, model order reduction, or choice of co-ordinates. The purpose of the paper is to motivate the reintroduction of geometric structure into systems theory. First, a brief introduction to the more common geometric structures is given, also showing their linear counterparts. It is then shown how these structures arise in systems theory by introducing the nonlinear control problems involved with mechanical manipulators, electrical networks, rotating electrical machinery and attitude control of spacecraft. The paper is concluded by considering the application of some of the geometric structures to nonlinear Hamiltonian and potential input/output systems.