Abstract: This article was based on the author’s reflections on the new ways of teaching geometry in school associated with computer technologies. We face the need for radical change to the traditional way of teaching geometry. We came to the understanding of this fact from analyzing the new trends in geometry research and the new demands on mathematics education in the present state of the art. The traditional way of teaching geometry in Russian school which is based on Euclid’s “Origins” has failed to keep up with the new demands. The future progress in geometry education will depend greatly on international cooperation and sharing of new experiences in the teaching of the subject. In this paper I shall focus on some experiences which are associated with the problems of computer analysis and control of the pupils’ activities while solving geometrical problem and with the geometry training of advanced pupils..