The book is a new edition of a volume published by Academic Press under the same title in 1978. An appendix entitled ‘Some Details’ (consisting of 13 pages of clarifying footnotes) has been added. We recall the content of the book for the younger generation: Elementary Differential Geometry, Lie Groups and Lie Algebras. Structure of Semisimple Lie Algebras, Symmetric Spaces, Decomposition of Symmetric Spaces, Symmetric Spaces of the Noncompact type, Symmetric Spaces of the Compact Type. Hermitian Symmetric Spaces, Structure of Semisimple Lie groups, The Classification of Simple Lie Algebras and Symmetric Spaces.