Abstract: In this talk I will explain how Legendrian contact homology can be used to produce positive lower bounds for the topological entropy of Reeb flows on contact 3-manifolds. As an application I will explain a contact topological generalization of a result of Denvir and Mackay which says that if a Riemannian metric on the two-dimensional torus has a simple contractible closed geodesic then its geodesic flow has positive topological entropy. This is partly joint work with Pedro A. S. Salomão.