In this talk, we start by reviewing recent results on the dynamics of Reeb vector fields defined by contact forms on three-dimensional manifolds, and then introduce Reeb fields defined by stable Hamiltonian structures. These are more general and arise, for instance, in stable regular energy level sets of Hamiltonian systems. We give a characterization of Reeb fields that are aperiodic or that have finitely many periodic orbits (under a certain nondegeneracy assumption). Finally, we give sufficient conditions for the existence of an adapted broken book decomposition or the existence of a Birkhoff section. This is joint work with A. Rechtman.