This book is a guided tour of geometry, from Euclid through to algebraic geometry. It shows how mathematicians use a variety of techniques to tackle problems, and it links geometry to other branches of mathematic.s It is a teaching text, with a large number of exercises woven into the exposition. Topics covered: ruler and compasses constructions, transformations, triangle and circle theorems, classification of isometries and groups of isometries in dimensions 2 and 3, Platonic solids, conics, similarities, affine, projective and Mobius transformations, non-Euclidean geometry, projective geometry, the beginnings of algebraic geometry.