Theory of groups and symmetries : finite groups Lie groups and Lie algebras Groups and transformations Lie groups Lie algebras Representations of groups and Lie algebras Compact Lie algebras Root systems and classification of simple Lie algebras Homogeneous spaces and their geometry Solutions to selected problems The book presents the main approaches in study of algebraic structures of symmetries in models of theoretical and mathematical physics namely groups and Lie algebras and their deformations. It covers the commonly encountered quantum groups (including Yangians). The second main goal of the book is to present a differential geometry of coset spaces that is actively used in investigations of models of quantum field theory gravity and statistical physics. The third goal is to explain the main ideas about the theory of conformal symmetries which is the basis of the AdS CFT correspondence. The theory of groups and symmetries is an important part of theoretical physics. In elementary particle physics cosmology and related fields the key role is played by Lie groups and algebras corresponding to continuous symmetries. For example relativistic physics is based on the Lorentz and Poincare groups and the modern theory of elementary particles the Standard Model is based on gauge (local) symmetry with the gauge group SU(3) x SU(2) x U(1). This book presents constructions and results of a general nature along with numerous concrete examples that have direct applications in modern theoretical and mathematical physics