A commentary on Teichm\"uller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale"

We provide a commentary on Teichm{\"u}ller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (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 rediscovered later on and published without any reference to Teichm{\"u}ller.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.The final version of this paper, together with the English translation of Teichm{\"u}ller's paper, will apper in Volume V of the \emph{Handbook of Teichm{\"u}ller theory} (European Mathematical Society Publishing House, 2015).