This is a revised edition of the ?rst printing which appeared in 2002. The book is based on lectures at the University of Bergen, Norway.Over the years these lectures have covered many different aspects and facets of the wonderful ?eld of geometry. Consequently it has never been possible to give a full and ?nal account of geometry as such, at an undergraduatelevel: A carefully consideredselection has always been necessary. The present book constitutes the main central themes of these selections. One of the groups I am aiming at, is future teachers of mathematics. All too oftenthe textsdealingwith geometrywhichgo intothe syllabusforteacher-students present the material in ways which appear pedantic and formalistic, suppressing the very powerful and dynamic character of this old ?eld, which at the same time so young. Geometry is a ?eld of mathematical insight, research, history and source of artistic inspiration. And not least important, an integral part of our common cultural heritage.