Up to minor changes, these are the notes of a series of lectures given at the Tata Institute of Fundamental Research, Mumbai in 1961. They cover some basic material on affine connections, locally or globally Riemannian and Hermitian symmetric spaces. A 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.