This book is a translation of a Russian text. The translation is exact, and the language used by the author has not been brought up to date. The volume is probably most useful as a source of supplementary materials for high school mathematics. It is also useful for teachers to broaden their mathematical background. Chapters included in the text are: (1) Geometric Figures as Point Sets; (2) Geometric Constructions; (3) The Transformation of Figures; (4) Parallel Translations; (5) Rotation; (6) Symmetry; (7) Similarity; (8) Inversion; (9) The General Problem of Measuring Lengths, Areas, and Volumes; (10) Euclid's "Elements"; (11) The Geometry of Lobachevskii; (12) The Axiomatic Structure of Geometry; and (13) The Idea of an Interpretation of a Geometric System. A selected bibliography is included. (RH)