We prove that if a geodesic flow on a surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature, as well as it applies to other previously unknown contexts such as the ones constructed by Donnay. The proof uses the existence of Birkhoff sections (Alves-Mazzucchelli) and of irreducible codings of homoclinic classes of measures (Buzzi-Crovisier-Lima). This is a joint work with Davi Obata and Mauricio Poletti.