This article, originally circulated as a preprint about 12 years ago, discusses the classification of diffeomorphisms of surfaces and the topology of surfaces. The author outlines the development of this theory, including its connections to Riemann surfaces, quasiconformal maps, Teichmüller theory, and Nielsen's theory of dynamical behavior of surfaces at infinity. The article also covers the classification of surface automorphisms from the perspective of measured foliations and the work of Lipman Bers, Dennis Sullivan, and others. The theory of measured laminations and 2-dimensional train tracks in three dimensions is explored, along with its applications to the classification of incompressible surfaces in 3-manifolds. The article concludes with a discussion of recent developments, such as the classification of automorphisms of free groups and the work of Kerckhoff, Bers, and others on the boundaries of Teichmüller space. The author emphasizes the geometric and intuitive aspects of the theory, highlighting the concrete nature of the induced transformations and the natural geometric structure on the space of measured foliations.This article, originally circulated as a preprint about 12 years ago, discusses the classification of diffeomorphisms of surfaces and the topology of surfaces. The author outlines the development of this theory, including its connections to Riemann surfaces, quasiconformal maps, Teichmüller theory, and Nielsen's theory of dynamical behavior of surfaces at infinity. The article also covers the classification of surface automorphisms from the perspective of measured foliations and the work of Lipman Bers, Dennis Sullivan, and others. The theory of measured laminations and 2-dimensional train tracks in three dimensions is explored, along with its applications to the classification of incompressible surfaces in 3-manifolds. The article concludes with a discussion of recent developments, such as the classification of automorphisms of free groups and the work of Kerckhoff, Bers, and others on the boundaries of Teichmüller space. The author emphasizes the geometric and intuitive aspects of the theory, highlighting the concrete nature of the induced transformations and the natural geometric structure on the space of measured foliations.