The book gives an introduction to topology at the advanced undergraduate to beginning graduate level, with an emphasis on its geometric aspects. Part I contains three chapters on basic point set topology, classification of surfaces via handle decompositions, and the fundamental group appropriate for a one semester or two quarter course. Carefully developed exercise sets support high student involvement. Besides exercises embedded within a chapter, there are extensive supplementary exercises to extend the material. Surfaces occur as key examples in treatments of the fundamental group, covering spaces, CW complexes, and homology in the last four chapters. Each chapter of Part I ends with a substantial project. Part II is written in a problem based format. These problems contain appropriate hints and background material to enable the student to work through the basic theory of covering spaces, CW complexes, and homology with the instructor's guidance. Low dimensional cases provide motivation and examples for the general development, with an emphasis on treating geometric ideas first encountered in Part I such as orientation. Part II allows the book to be used for a year long course at the first year graduate level.The book's collection of over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix. Solutions to all exercises are available to the instructor in electronic form.This text forms the latest in the Oxford Graduate Texts in Mathematics series which publishes textbooks suitable for graduate students in mathematics and its applications. The level of books may range from some suitable for advanced undergraduate courses at one end, to others of interest to research workers. The emphasis is on texts of high mathematical quality in active areas, particularly areas that are not well represented in the literature at present.