The theory of embedded graphs (i.e. graphs drawn on two-dimensional surfaces) always attracted researchers by its beauty and by a large variety of difficult combinatorial and geometric questions. For a long time this theory looked rather isolated. But the last few decades witnessed an appearance of entirely unexpected new applications of embedded graphs, ranging from Galois theory to quantum gravity models. The theory has suddenly become a kind of a nervous center of a vast field of research. The book provides an accessible introduction to this new domain, touching such topics as ramified coverings of Riemann surfaces, Galois group action on embedded graphs (the Grothendieck theory of 'dessins d'enfants'), matrix integrals method for map enumeration, moduli spaces of complex corves, topological classification of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants. The presentation is as concrete as possible, with numerous figures, examples (including computer calculations) and exercises, and will be of great interest to graduate students and researchers.