This is a book of elementary geometric topology, in which geometry, frequently illustrated, guides calculation. The book starts with a wealth of examples, often subtle, of how to be mathematically certain whether two objects are the same from the point of view of topology. After introducing surfaces, such as the Klein bottle, the book explores the properties of polyhedra drawn on these surfaces. Even in the simplest case, of spherical polyhedra, there are good questions to be asked. There are many examples and exercises making this a useful textbook for a first undergraduate course in topology. For much of the book the prerequisites are slight, though, so anyone with curiosity and tenacity will be able to enjoy the book. As well as arousing curiosity, the book gives a firm geometrical foundation for further study. "A Topological Aperitif provides a marvellous introduction to the subject, with many different tastes of ideas.Stephen Huggett and David Jordan have excellent credentials for explaining the beauty of this curiously austere but potentially enormously general form of geometry". Professor Sir Roger Penrose OM FRS, Mathematical Institute, Oxford, UK