Geometry is central to many branches of mathematics and physics, and offers a whole range of views on the universe. This is an introduction to the ideas and includes generous helpings of simple explanations and examples. Prerequisites are minimal, so it is ideally suited to beginning courses in, and newcomers to, the subject. It lays special emphasis on coordinate systems and the coordinate changes that generate symmetries. The discussion progresses from Euclidean to non-Euclidean geometries (including spherical and hyperbolic geometry), through the group-theoretic approach to symmetries to topological ideas, such as the Möbius strip, the Klein bottle and surfaces with handles. The geometry of transformation groups emerges, and is applied to relativity theory and quantum mechanics. The book is based in many years teaching experience so is thoroughly class-tested. There are plenty of illustrations; chapters end with a collection of exercises, solutions to which are available for teachers.