Geometry is central to many branches of mathematics and physics, and offers a complete range of views on the universe. This introduction includes many simple explanations and examples. With minimal prerequisites, the book provides a first glimpse of many research topics in modern algebra, geometry and theoretical physics. The book is based on many years' teaching experience, and is thoroughly class-tested. There are copious illustrations, and each chapter ends with exercises. Further teaching material is available via the web, including assignable problem sheets with solutions.