  The type problem is the problem of deciding, for a simply connected Riemann
surface, whether it is conformally equivalent to the complex plane or to the
unit dic in the complex plane. We report on Teichm{\"u}ller's results on the
type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf
das Typenproblem (An application of quasiconformal map-pings to the type
problem) (1937) and Untersuchungen{\"u}ber kon-forme und quasikonforme
Abbildungen (Investigations on con-formal and quasiconformal mappings) (1938).
They concern simply connected Riemann surfaces defined as branched covers of
the sphere. At the same time, we review the theory of line complexes, a
combinatorial device used by Teichm{\"u}ller and others to encode branched
coverings of the sphere. In the first paper, Teichm{\"u}ller proves that any
two simply connected Riemann surfaces which are branched coverings of the
Riemann sphere with finitely many branch values and which have the same line
complex are quasiconformally equivalent. For this purpose, he introduces a
technique for piecing together quasi-conformal mappings. He also obtains a
result on the extension of smooth orientation-preserving diffeomorphisms of the
circle to quasiconformal mappings of the disc which are conformal at the
boundary. In the second paper, using line complexes, Teichm{\"u}ller gives a
type criterion for a simply-connected surface which is a branched covering of
the sphere, in terms of an adequately defined measure of ramification, defined
by a limiting process. The result says that if the surface is ''sufficiently
ramified'' (in a sense to be made precise), then it is hyperbolic. In the same
paper, Te-ichm{\"u}ller answers by the negative a conjecture made by
Nevan-linna which states a criterion for parabolicity in terms of the value of
a (different) measure of ramification, defined by a limiting process.
Teichm{\"u}ller's results in his first paper are used in the proof of the
results 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 Society
Publishing House, 2020).
