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

A Commentary on Teichm{\"u}ller's paper "Untersuchungen \"uber konforme  und quasikonforme Abbildungen" (Investigations on conformal and  quasiconformal mappings) (to appear in Vol. VII of the \emph{Handbook of  Teichm\"uller theory}