This book, based on lectures given by the author in the first year of the Independent Moscow University, is an introduction to Euclidean, spherical, projective and hyperbolic (Lobachevsky) geometry, written in a synthetic, coordinate-free style; along the way, the reader also gets acquainted with the beginnings of group theory and finds out in connection with which this theory arose. The book is equipped with a lot of exercises to help you master the material.