Graph theory is one of the fastest-growing branches of mathematics. Until recently, it was regarded as a branch of combinatorics and was best known by the famous four-colour theorem stating that any map can be coloured using only four colours such that no two bordering countries have the same colour. Now graph theory is an area of its own. Graph theory has numerous applications in almost every field of science and has attracted new interest because of its relevance to such technological problems as computer and telephone networking and, of course, the Internet. In this book in the "Johns Hopkins Studies in the Mathematical Science" series, Bojan Mohar and Carsten Thomassen look at a relatively new area of graph theory: that associated with curved surfaces. Graphs on surfaces form a natural link between discrete and continuous mathematics. The book provides a rigorous and concise introduction to graphs on surfaces and surveys some of the recent developments in this area. Among the basic results discussed are Kuratowski's theorem and other planarity criteria, the Jordan Curve Theorem and some of its extensions, the classification of surfaces, and the Heffter-Edmonds-Ringel rotation principle, which makes it possible to treat graphs on surfaces in a purely combinatorial way. The genus of a graph, contractability of cycles, edge-width, and face-width are treated purely combinatorially, and several results related to these concepts are included. The extension by Robertson and Seymour of Kuratowski's theorem to higher surfaces is discussed in detail, and a shorter proof is presented. The book concludes with a survey of recent developments on colouring graphs on surfaces.