This text examines the topology of compact surfaces through the development of some simple ideas in place geometry, an approach which allows a treatment of an area of mathematics that is particularly attractive both for the richness of its applications, and for the variety of its interactions with other branches of mathematics. A wide variety of topics are linked with surface topology, such as graph theory, group theory, vector field theory and plane Euclidean and non-Euclidean geometry. Each topic is treated from its beginnings, thus offering a bird's eye view of many branches of mathematics. The author's introductions are presented in simple terms and, for those who wish to extend their knowledge of the topics discussed, a bibliography is provided.