This book is intended as a text for a course in differential geometry, at the research level. Differential geometry has a long history as a field of mathematics and yet its rigorous foundations in the realm of contemporary mathematics in relativity. The theory of structures on manifolds is a very interesting topic of modern differential geometry and differential geometric aspects of submanifold of manifolds with certain structures are vast and very fruitful fields for Riemannian geometry. The purpose of this book is to provide an introduction to the theory of various differential geometric structures on manifolds. In all the chapters, the author has tried to familiarize the readers with various techniques of computations which are currently in use in differential geometry. These are; (a) classical tensor calculus (b) exterior differential calculus with E. Cartan ( c) formalism of covariant differentiation D_X Y.