This book is intended to serve as a text book for the present B. A. and B. Sc. Honours courses of Indian Universities, It consists of five chapters named chapters 0, I, II, III and IV of which chapter 0 provides an informative introduction concerning the origin and nature of the tensor concept and the scope of tensor calculus, while chapter I deals with some preliminaries necessary for the treatment of the material in the succeeding chapters. In Chapter II Tensor Algebra has been developed in an n-dimensional space, while in chapter III an n-dimensional Riemannian space has been chosen for the development of Tensor Calculus. In Chapter IV it is shown how some known results of ordinary vector calculus can be derived from appropriate results in the tensor calculus of a 3-dimensional Riemannian space. This is done to indicate the power of tensor calculus. Every effort has been made to treat the subject matter in an unambiguously clear and rigorous manner. To ensure clarity of understanding detailed explanations have been given wherever thought necessary. The numerous notes to be found in the text are designed to help the reader to have a firm grasp on the material and are also aimed at preventing false impressions and distorted conceptions At the end of each chapter, except chapter 0, a large number of problems of various types have been completely solved and exercises containing carefully graded and motivating examples have been incorporated.