This book is designed as an introduction to the theory of Lie groups by means of matrix subgroups of the real or complex linear group. The main examples treated are the special linear groups, orthogonal and special orthogonal groups, unitary groups, symplectic groups and the Lorentz group. In particular, relations between complex matrix groups and real matrix groups are discussed. The author also treats various algebraic, analytic and topological properties (e.g., norms, metric, compactness and group actions). The second part contains a study of algebras, quaternions, quaternionic symplectic groups, Clifford algebras and spinor groups and their special properties. The third part can be considered as an introduction to the theory of Lie groups and homogeneous spaces. Classical examples are explained. The last part contains special topological and geometrical problems (connectivity of matrix groups, description of maximal tori in compact Lie groups, semi-simple factorisation, roots systems, Weyl groups and Dynkin diagrams). The book is a nice elementary introduction to the theory of Lie groups.