A more accurate title would be: Introduction to some aspects of the analytic theory of automorphic forms on SL2(ℝ) and the upper half-plane X. Originally, automorphic forms were holomorphic or meromorphic functions on X satisfying certain conditions with respect to a discrete group Γ of automorphisms of X, usually with fundamental domain of finite (hyperbolic) area. Later on, H. Maass – and then A. Selberg and W. Roelcke – dropped the assumption of holomorphicity, requiring instead that the functions under consideration be eigenfunctions of the Laplace–Beltrami operator. In the 1950s it was realized (in more general cases) – initially by I. M. Gelfand and S. V. Fomin, and then by Harish-Chandra – that the automorphic forms (holomorphic or not) could be equivalently viewed as functions on Γ\SL2(ℝ) satisfying certain conditions familiar in the theory of infinite dimensional representations of semisimple Lie groups. This led to a new outlook, where the Laplace–Beltrami operator is replaced by the Casimir operator and the theory of automorphic forms becomes closely related to harmonic analysis on Γ\SL2(ℝ). This is the point of view adopted in this presentation. However, in order to limit the prerequisites, no knowledge of representation theory is assumed until the last sections, a main purpose of which is precisely to make this connection explicit. A fundamental role is played throughout by a theorem stating that a function on SL2(ℝ) satisfying certain assumptions (Z-finite and K-finite) is fixed under convolution by some smooth function with arbitrarily small compact support around the identity element (2.14).