This book, Principles of Mathematical Analysis, is designed to meet the requirements of the course in Mathematical Analysis of the students of B.Sc. (Hons.) and M.Sc. of Indian Universities. The examinees who are preparing for the competitive examination will also find it very useful. The aim has been to provide the latest developments in the subject in an honest, rigorous, up-to-date manner, and at the same time not too pedantic. The book provides a transition from elementary calculus to advanced course in real analysis, and it introduces the reader to some of the abstract concepts that pervade modern analysis. At the end of the each section a set of problems has been given to illustrate definitions and theorems. Numerical examples are provided which are very important from the theoretical concepts, as well as, examination viewpoint. 1 sets and functions. 2 dedikind's theory of real number system. 3 cantor's theory of real numbers. 4 basic topology. 5 sequences of real numbers. 6 infinite series. 7 limits and continuity of functions. 8 defferentiation. 9 the riemann integral. 10 the riemann stietjes integral. 11 improper integrals. 12 sequence and series of functions. 13 basic functions 14 power series. 15 fourier series. 16 functions of several variables. 17 implicit function. 18 integration on r2 line integrals, double integrals. 19 integration on r2 line, surface and volume integrals. 20 the lebesgue integral and measure. 21 gamma and beta function.