These notes are based on lectures given by Jussi Väisälä at the University of Helsinki in 1967-1968. They were initially intended for publication in another series but were transferred to Springer-Verlag in 1971 due to delays. The author made minor changes and added references to the latest literature. The notes aim to provide an exposition of the basic theory of quasiconformal mappings in R^n. The books of Ahlfors and Lehto-Virtanen provide historical background, though no prior knowledge of 2-dimensional quasiconformal mappings is required. The proofs apply to n=2, but are often simplified due to the Riemann mapping theorem. The reader is assumed to be familiar with basic facts of measure and integration, with more advanced results in Chapter 3. The text also assumes knowledge of the topology of Euclidean spaces. Two important topics are omitted: the theorem of Gehring and Rešetnjak and Gehring's theory on the symmetrization of rings. The quasiconformal mappings form a subclass of quasiregular mappings, which are not necessarily homeomorphisms. The notes include references and brief historical remarks at the end of each section, with a comprehensive bibliography in Caraman's monograph. The text defines notation and terminology, including sets, spaces, and functions, and discusses the properties of quasiconformal mappings, their definitions, and applications. The notes cover the modulus of a curve family, quasiconformal mappings, real analysis background, analytic properties of quasiconformal mappings, and mapping problems. The text is structured into chapters and sections, with detailed definitions and examples.These notes are based on lectures given by Jussi Väisälä at the University of Helsinki in 1967-1968. They were initially intended for publication in another series but were transferred to Springer-Verlag in 1971 due to delays. The author made minor changes and added references to the latest literature. The notes aim to provide an exposition of the basic theory of quasiconformal mappings in R^n. The books of Ahlfors and Lehto-Virtanen provide historical background, though no prior knowledge of 2-dimensional quasiconformal mappings is required. The proofs apply to n=2, but are often simplified due to the Riemann mapping theorem. The reader is assumed to be familiar with basic facts of measure and integration, with more advanced results in Chapter 3. The text also assumes knowledge of the topology of Euclidean spaces. Two important topics are omitted: the theorem of Gehring and Rešetnjak and Gehring's theory on the symmetrization of rings. The quasiconformal mappings form a subclass of quasiregular mappings, which are not necessarily homeomorphisms. The notes include references and brief historical remarks at the end of each section, with a comprehensive bibliography in Caraman's monograph. The text defines notation and terminology, including sets, spaces, and functions, and discusses the properties of quasiconformal mappings, their definitions, and applications. The notes cover the modulus of a curve family, quasiconformal mappings, real analysis background, analytic properties of quasiconformal mappings, and mapping problems. The text is structured into chapters and sections, with detailed definitions and examples.