  This is a commentary on Teichm{\"u}ller's paper Unter-suchungen{\"u}ber
konforme und quasikonforme Abbildungen (Inves-tigations on conformal and
quasiconformal mappings) published in 1938. The paper contains fundamental
results in conformal geometry , in particular a lemma, known as the Modulsatz,
which insures the almost circularity of certain loci defined as complementary
components of simply connected regions in the Riemann sphere, and another
lemma, which we call the Main Lemma, which insures the circularity near
infinity of the image of circles by a qua-siconformal map. The two results find
wide applications in value distribution theory, where they allow the efficient
use of moduli of doubly connected domains and of quasiconformal mappings.
Te-ichm{\"u}ller's paper also contains a thorough development of the theory of
conformal invariants of doubly connected domains.The final version of this
paper will appear in Vol. VII of the \emph{Handbook of Teichm{\"u}ller theory}
(European Mathematical Society Publishing House, 2020).

Complex Variables;Geometric Topology;History and Overview