There is an intimate interplay between certain classes of dynamical systems and holomorphic curve theory. For Reeb vector fields on closed three-dimensional manifolds this relationship gives a tool to construct global surfaces of section and generalizations thereof. The class of Reeb vector fields is quite large. For example classical Hamiltonian systems, restricted to a regular energy surface, are described by Reeb vector fields. As an application one obtains for example the result that on a strictly convex energy surface in a four-dimensional symplectic vector space there are either precisely two geometrically distinct periodic orbits or infinitely many. Other applications will also be described.