This book provides a broad foundation for topology, including general, geometric, differential, combinatorial and algebraic topology. It is designed for use either for a first course in topology beginning in chapter one, a first course in geometric topology beginning in chapter three or a course in applied topology. The book is structured around the theoretical framework of topology, and abundant illustrations and applications provide intuition and put the subject in modern setting. Topics include open sets, compactness, homotopy, surface classification, index theory on surfaces, manifolds and complexes, topological groups, the fundamental group and homology. Modern applications of topology have played an important role solving a diverse spectrum of applied problems. In this text serious attention is given to recent applications of topology in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, population modeling and other areas of science and engineering. Most applications are presented in optional sections, allowing an instructor to customize the presentation.