Regular polytopes and their symmetry have a long history stretching back two and a half millennia, to the classical regular polygons and polyhedra.Much of modern research focuses on abstract regular polytopes, but significant recent developments have been made on the geometric side, including the exploration of new topics such as realizations and rigidity, which offer a different way of understanding the geometric and combinatorial symmetry of polytopes.This is the first comprehensive account of the modern geometric theory, and includes a wide range of applications, along with new techniques.While the author explores the subject in depth, his elementary approach to traditional areas such as finite reflexion groups makes this book suitable for beginning graduate students as well as more experienced researchers.