I. Introduction.- II. Some Mathematical Preliminaries.- III. Ito Integrals.- IV. Stochastic Integrals and the Ito Formula.- V. Stochastic Differential Equations.- VI. The Filtering Problem.- VII. Diffusions: Basic Properties.- VIII. Other Topics in Diffusion Theory.- IX. Applications to Boundary Value Problems.- X. Application to Optimal Stopping.- XI Application to Stochastic Control.- Appendix A: Normal Random Variables.- Appendix B: Conditional Expectations.- Appendix C: Uniform Integrability and Martingale Convergence.- List of Frequently Used Notation and Symbols.