This is a pleasant and reasonable short and non-technical introduction to the topology and geometry of 2- and 3-dimensional manifolds, recommendable to everyone (in particular, non-specialists) who wants to learn about basic facts and ideas as well as some recent developments in this growing field, which has seen such an enormous progress in the last 20 years developing connections with various other fields of mathematics. First, the classification and recognition problem for 2-manifolds is discussed, with the Euler characteristic as the basic invariant, as well as hyperbolic 2-manifolds. The paper then passes to 3-manifolds, especially hyperbolic 3-manifolds and the effectiveness of their invariants: homology, volume, the Chern-Simons and the \(\eta\)-invariant. Here the article becomes an interesting mixture between basic constructions and recent results and problems.