This book grew out of courses given at the University of Wales Swansea to second- and third-year undergraduates. It is designed to provide enough material for a one-year course and splits naturally into a preliminary topology course and a follow-on course in algebraic topology. It is often said that topology is a subject which is poorly served for textbooks, and when preparing the lecture courses I found no book that was both accessible to our undergraduates and relevant to current research in the field. This book is an attempt to fill that gap. It is generally accepted that a oneyear course on topology is not long enough to take a student to a level where she or he can begin to do research, but I have tried to achieve that as nearly as possible.