The Reeb flow of a contact form is a generalisation of Hamiltonian flows on energy hypersurfaces in classical mechanics. In this talk I shall address the question of how "complicated" such flows can be. Among other things, I plan to discuss a construction of Reeb flows with a global surface of section on which the Poincaré return map is a pseudorotation. This is joint work with Peter Albers and Kai Zehmisch