We give a self-contained proof of the Theorem of Fathi and Shub on the entropy minimizing property of pseudo-Anosov self-homeomorphisms of closed Riemann surfaces of genus at least two following mainly along the lines of the original proof. A proof of the theorem for the case of punctured surfaces is also included. The exposition contains a thorough account on the necessary prerequisites on the trajectories of quadratic differentials. The present proof of the Fathi and Shub Theorem on the relation between the entropy and the quasi-conformal dilatation is new. The chapter concludes with a thorough treatment of the entropy of self-homeomorphisms of Riemann surfaces of second kind, which is a prerequisite for the treatment of reducible braids.