The book provides a systematic presentation of various geometries - Euclidean, affine, projective, elliptical, hyperbolic, infinite-dimensional. Problems of different geometries are considered from a single point of view, and everywhere there are common roots of different phenomena. All geometric objects are examined from the standpoint of duality. The theory of grasshoppers and quadrics is detailed, including the theory of grasshoppers for non-Euclidean geometries. The book contains many vivid geometric facts, solved many beautiful geometric problems. Numerous drawings help to more clearly imagine the geometric theorems presented . At the end of the chapters are tasks and exercises that allow you to use the book as a textbook. The book is intended to contribute to the development of geometric research and the improvement of mathematical education. For schoolchildren, students, teachers of mathematics . The first edition of the book was published in 1997.