This book presents the previously unpublished notes from a series of lectures given by the author at the Tata Institute of Fundamental Research in 1961. Basic material on affine connections and on locally or globally Riemannian and Hermitian symmetric spaces is covered. The final chapter proves the basic theorems on maximal compact subgroups of Lie groups. Readers should be familiar with differential manifolds and the elementary theory of Lie groups and Lie algebras.