The book deals with the geometric interpretation of all simple Lie groups in the form of groups of movements of the classical non-Euclidean geometries of Lobachevsky and Riemann, the main groups of projective, conformal, symplectic and metasymplectic geometries over algebras. The book also discusses the geometric interpretation of Lie groups obtained by marginal transitions from simple Lie groups. Such groups include groups of motions of Euclidean, pseudo-Euclidean, isotropic and many other geometries over algebras. Along with continuous spaces, finite spaces are considered, the main groups of which are finite Lie-type groups. The book indicates the most important applications of the geometries under consideration to physics. The book is designed for researchers, graduate students and students specializing in non-Euclidean geometries .