The main aim of this talk is to present the complex hyperbolic space, and give many reasons why people from different mathematical horizons could be interested in it. I will start with a nice non-technical description of the "Poincaré disk" borrowed from Poincaré's book La Science et l'Hypothèse. Then I will discuss some features of the higher-dimensional complex hyperbolic space. Depending on the time, I may also speak about the theory of holomorphic functions on quotients of this space.