Every compact surface admits a constant curvature metric, endowing it with a geometry. For surfaces of genus at least 2, this leads to hyperbolic geometry. The model space is the hyperbolic plane. We give a brief introduction to the beautiful world of hyperbolic geometry in dimension 2 starting from the hyperbolic plane and its isometries and then discussing hyperbolic surfaces. References for an extended treatment include, for example, Thurston (Collected works of William P. Thurston with commentary: IV. The geometry and topology of three-manifolds: with a preface by Steven P. Kerckhoff, 2022 ) or Benedetti and Petronio (Lectures on hyperbolic geometry, 1992); Marden (Hyperbolic manifolds. An introduction in 2 and 3 dimensions, 2016); Ramsay and Richtmyer (Introduction to hyperbolic geometry, 1995).