  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).
