The objective of this project is to study the topological entropy for Reeb flows and its relations with symplectic and contact topology. Reeb flows are a special class of dynamical systems which lie at the intersection of geometry, topology and mathematical physics. The class of Reeb flows includes the geodesic flows of Riemannian metrics and important examples of Hamiltonian dynamical systems. The dynamical properties of Reeb flows are strongly related to the topological properties of contact and symplectic manifolds. In this project we study the behaviour of the topological entropy of Reeb flows. The topological entropy is an important dynamical invariant which codifies in a single non-negative number the exponential complexity of a dynamical system. If the topological entropy of a dynamical system is positive then the system exhibits some type of chaotic behaviour. In this project we propose to: A) better understand the dynamics of Reeb flows with positive topological entropy using invariants coming from Floer theory; B) to use topological methods to construct new examples of Reeb flows with zero topological entropy; C) to use Floer theory to study how topological entropy varies under perturbations of Reeb flows.