A Commentary on Teichm\"uller's paper "\"Uber Extremalprobleme der konformen Geometrie" (On extremal problems in conformal geometry)

We comment on the paper \"Uber Extremalprobleme der konformen Geometrie (On extremal problems in conformal geometry) by Teichm\"uller, published in 1941. This paper contains ideas on a wide generalization of his previous work on the solution of extremal problems in conformal geometry. The generalization concerns at the same time the fields of function theory, topology and algebra. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm\"uller theory. This volume is dedicated to the memory of Alexander Grothendieck.