The book presents the theory of geometries, in small enough parts that coincide with the Euclidean .. A number of examples of such geometries are analyzed and their general theory is based on the concept of a uniformly discontinuous group of transformations. It describes the application of these concepts to crystallography, as well as their relationship with Lobachevsky's geometry. A purely geometric presentation does not require any knowledge beyond the high school mathematics program. For teachers of mathematical faculties of universities and pedestals . Can be used by students of relevant specialties and high school mathematics teachers.