Stochastic Analysis and Diffusion Processes presents a simple‎,‎ mathematical introduction to Stochastic Calculus and its applications‎.‎ The book builds the basic theory and offers a careful account of important research directions in Stochastic Analysis‎.‎ The breadth and power of Stochastic Analysis‎,‎ and probabilistic behavior of diffusion processes are told without compromising on the mathematical details‎.‎ Starting with the construction of stochastic processes‎,‎ the book introduces Brownian motion and martingales‎.‎ The book proceeds to construct stochastic integrals‎,‎ establish the Ito formula‎,‎ and discuss its applications‎.‎ Next‎,‎ attention is focused on stochastic differential equations ‎(‎SDEs‎)‎ which arise in modeling physical phenomena‎,‎ perturbed by random forces‎.‎ Diffusion processes are solutions of SDEs and form the main theme of this book‎.‎ The Stroock‎-Varadhan martingale problem‎,‎ the connection between diffusion processes and partial differential equations‎,‎ Gaussian solutions of SDEs‎,‎ and Markov processes with jumps are presented in successive chapters‎.‎ The book culminates with a careful treatment of important research topics such as invariant measures‎,‎ ergodic behavior‎,‎ and large deviation principle for diffusions‎.‎ Examples are given throughout the book to illustrate concepts and results‎.‎ In addition‎,‎ exercises are given at the end of each chapter that will help the reader to understand the concepts better‎.‎ The book is written for graduate students‎,‎ young researchers and applied scientists who are interested in stochastic processes and their applications‎.‎ The reader is assumed to be familiar with probability theory at graduate level‎.‎ The book can be used as a text for a graduate course on Stochastic Analysis‎.‎