We provide a mathematical commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale<(i> (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.- Naturw. Kl. 1940, No.22, 1–197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were re- discovered later on and published without any reference to Teichmüller. In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.