Provides a rigorous introduction to the modern theory of stochastic calculus, with a particular emphasis on continuous time, continuous path stochastic processes; the canonical example being Brownian motion. The main topics covered include basic definitions relating to stochastic processes: filtrations; measurability; stopping times; cadlag processes; Martingales; quadratic variation, etc.; the fundamental inequalities, convergence results, optional sampling theorem, and other properties of Martingales; the Doob-Meyer decomposition; construction of local Martingales; the analysis of the space of continuous square integrable Martingales; the definition and construction of Brownian motion: the Kolmogorov-Daniell consistency theorem, the Kolmogorov-Centsov theorem on continuous modifications; the construction of Wiener measure; analysis of Brownian motion: Markov, strong Markov properties; path regularity and distributional properties; the construction of the stochastic integral with respect to a continuous Martingale; Ito's change of variable formula; representation results; Girsanov's theorem; local time of Brownian motion; and Stochastic differential equations and diffusions: construction of strong and weak solutions; the Martingale problem; connections with partial differential equations and harmonic analysis; the Feynman-Kac formula.