The book by Gorbatsevich, Onishchik and Vinberg is the first in a series of volumes devoted to the theory of Lie groups and Lie algebras. The first part of the book deals with the foundations of the theory based on the classical global approach of Chevalley followed by an exposition of the alternative approach via the universal algebra and the Campbell-Hausdorff formula. It also contains a survey of certain generalizations of Lie groups. The second more advanced part treats the topic of Lie transformation groups covering e.g. properties of orbits and stabilizers, homogeneous fibre bundles, Frobenius duality, groups of automorphisms of geometric structures, Lie algebras of vector fields and the existence of slices. The work of the last decades including the most recent research results is covered. The book contains numerous examples and describes connections with topology, differential geometry, analysis and applications. It will be of great interest to graduate students and researchers in mathematics and theoretical physics.

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