Almost everyone is acquainted with plane Euclidean geometry as it is usually taught in high school. This book introduces the reader to a completely different way of looking at familiar geometrical facts. It is concerned with transformations of the plane that do not alter the shapes and sizes of geometric figures. Such transformations play a fundamental role in the group theoretic approach to geometry. The treatment is direct and simple. The reader is introduced to new ideas and then is urged to solve problems using these ideas. The problems form an essential part of this book and the solutions are given in detail in the second half of the book.