Vasiliev, A. Yu.; Sergeev, A. G. Classical and quantum Teichmüller spaces. (English. Russian original) Zbl 1285.30026 Russ. Math. Surv. 68, No. 3, 435-502 (2013); translation from Usp. Mat. Nauk 68, No. 3, 39-110 (2013). Teichmüller theory has many connections with other directions in mathematical sciences, and is closely related to mathematical physics. In this survey the authors present the main directions of the development of Teichmüller theory and its applications to string theory. In the first chapter they introduces the protagonists of Teichmüller theory: Teichmüller, Ahlfors and Bers. The other two chapters are devoted to the developments related to the Teichmüller space of compact Riemann surfaces and the universal Teichmüller space, respectively, containing basic theory and recent results.After introducing the definition of the Teichmüller space of finite Riemann surfaces and its tangent and cotangent spaces, the authors are particularly interested in the Kobayashi metric and the Carathéodory metric. They introduce several non-equivalent methods for compactifying the Teichmüller space. In the eighth subsection of Chapter 2, the authors discuss the harmonic properties of a functional called the modulus on the Teichmüller space and describe the Teichmüller metric in terms of it.In Chapter 3, the authors present some main properties of the universal Teichmüller space \(\mathcal{T}\), such as its complex structures, the tangent map of the composite map of the natural projection with the Bers embedding, the Kähler metric on \(\mathcal{T}\). They discuss some subspaces of the universal Teichmüller space, such as the classical Teichmüller space \(T(G)\), the space of normalized diffeomorphisms. In the fifth subsection of Chapter 3, they focus on a Grassmannian realization of the universal Teichmüller space. The last two subsections are devoted to the geometric quantizations of the space of normalized diffeomorphisms and the universal Teichmüller space. Reviewer: Chong Wu (Chengdu) Cited in 2 Documents MSC: 30F60 Teichmüller theory for Riemann surfaces 30C62 Quasiconformal mappings in the complex plane 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 81S10 Geometry and quantization, symplectic methods 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30F10 Compact Riemann surfaces and uniformization 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) Keywords:Teichmüller space; quasiconformal map; universal Teichmüller space; quasisymmetric homeomorphism; Beltrami differential; geometric quantization PDFBibTeX XMLCite \textit{A. Yu. Vasiliev} and \textit{A. G. Sergeev}, Russ. Math. Surv. 68, No. 3, 435--502 (2013; Zbl 1285.30026); translation from Usp. Mat. 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