This book is an expanded version of a series of lectures given by one of the authors for a special semester-long program for advanced undergraduates. It provides an introduction to the geometry of surfaces, with a recurring theme of Euler characteristic as a guiding principle. After a warm-up chapter on various ways of constructing a surface, the authors set up for a series of journeys always ending with yet another manifestation of this invariant, including: triangulations on spaces, Betti numbers, Morse functions, the index of a vector field and the Gauss-Bonnet Theorem. The style of the book puts emphasis on motivation rather than clear-cut division into definitions, theorems and proofs. There are a lot of pictures and exercises with hints provided for most of them. The book can be recommended to anyone who wants to get acquainted with the most important ideas of geometry in a quick and elegant way.