A transverse foliation is a generalisation of an open book decomposition adapted to Reeb flow. It is a singular foliation whose singular set is a finite set of periodic orbits, called binding orbits, and regular leaves are surfaces asymptotic to the binding orbits and transverse to the flow. In this talk I will give a gentle introduction to transverse foliations and report some recent progress. It is based on joint work with de Paulo, Hryniewicz, and Salomão.