I will discuss some current joint work with Helmut Hofer in which we make use of symplectic topology and pseudoholmorphic curves to study properties of Hamiltonian flows on compact regular hypersurfaces of symplectic manifolds. In particular, I will show how pseudoholomorphic curve techniques can be used to prove that every non-empty, compact, regular energy surface in R^4 has a trajectory which is not dense in the energy level.