This is a gentle introduction to the problem of the classification of low-dimensional spaces and to the program of geometrization of two and 3-manifolds, both started more than a century ago with the Poincaré conjecture and Uniformization theorem, and completed some years ago by Thurston’s and Hamilton’s work and finally by the stunning results of Perelman. The beautiful results by Thurston showed the hitherto unexpected richness of geometry in three dimensions, and the great importance of the hyperbolic world. Perelman showed as the metric and the topology of 3-manifolds are profoundly related, and that the metric acts as a function of energy. In this process the Ricci flow play a key role. All these works offer a new landscape of mathematics and of the deep connections between topology, geometry and analysis. A few remarks will be devoted to sketch this new landscape and to highlight some applications especially in relativistic and quantum physics.