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The type problem is the problem of deciding, for a simply connected Riemannsurface, whether it is conformally equivalent to the complex plane or to theunit dic in the complex plane. We report on Teichm{\"u}ller's results on thetype problem from his two papers Eine Anwendung quasikonformer Abbildungen aufdas Typenproblem (An application of quasiconformal map-pings to the typeproblem) (1937) and Untersuchungen{\"u}ber kon-forme und quasikonformeAbbildungen (Investigations on con-formal and quasiconformal mappings) (1938).They concern simply connected Riemann surfaces defined as branched covers ofthe sphere. At the same time, we review the theory of line complexes, acombinatorial device used by Teichm{\"u}ller and others to encode branchedcoverings of the sphere. In the first paper, Teichm{\"u}ller proves that anytwo simply connected Riemann surfaces which are branched coverings of theRiemann sphere with finitely many branch values and which have the same linecomplex are quasiconformally equivalent. For this purpose, he introduces atechnique for piecing together quasi-conformal mappings. He also obtains aresult on the extension of smooth orientation-preserving diffeomorphisms of thecircle to quasiconformal mappings of the disc which are conformal at theboundary. In the second paper, using line complexes, Teichm{\"u}ller gives atype criterion for a simply-connected surface which is a branched covering ofthe sphere, in terms of an adequately defined measure of ramification, definedby a limiting process. The result says that if the surface is ''sufficientlyramified'' (in a sense to be made precise), then it is hyperbolic. In the samepaper, Te-ichm{\"u}ller answers by the negative a conjecture made byNevan-linna which states a criterion for parabolicity in terms of the value ofa (different) measure of ramification, defined by a limiting process.Teichm{\"u}ller's results in his first paper are used in the proof of theresults of the second one.The final version of this paper will appear in Vol.VII of the Handbook of Teichm{\"u}ller theory (European Mathematical SocietyPublishing House, 2020).

TeichmUller's work on the type problem (To appear in Vol. VII of the Handbook of Teichm{\"u}ller theory, 2020)