Except for some minor changes, these are the notes of a series of lectures given at the Tata Institute of Fundamental Research, Mumbai. They cover some basic material on affine connections, locally or globally Riemannian and Hermitian symmetric spaces. The last chapter proves the basic theorems on maximal compact subgroups of Lie groups with finitely many connected components. Familiarity with differential manifolds and the elementary theory of Lie groups and Lie algebras is assumed. A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels. Readership Graduate students and research mathematicians working in topological groups, Lie groups, and differential geometry. Reviews "The book is a careful development of the theory of Riemannian symmetric spaces."  Mathematical Reviews Table of Contents  Preliminaries
 Orthogonal involutive Lie algebras
 Locally symmetric spaces
 Riemannian symmetric spaces
 Compact groups. Klein forms of symmetric spaces
 Hermitian symmetric spaces
 Maximal compact subgroups of Lie groups
