In elementary Euclidean geometry, the congruent mappings play an important role already starting from the sixth grade. In the case of the composition (applying behind each other) the theorem about the mapping of three reflections is significantly. Within synthetic geometry it has been shown that (except for an assumption of richness) this theorem alone, taken as an axiom, is basis for all metric planes over a commutative field. This knowledge is already fifty years old, but it still has great significance today. This book is the translated, updated and expanded version of a German lecture note from the 1960s.The book is aimed at interested mathematicians as well as students of mathematics. In particular, it is suitable for teachers and teacher students of grammar schools and high schools as a mathematical background of geometry teaching concerning congruent mappings.