This course develops the basic tools from measure-theoretic probability theory and stochastic calculus that are needed for an in-depth study of continuous time finance. Some related tools from asset pricing (e.g., risk-preferences and state-price densities) are introduced as well, and the basic ingredients of continuous time financial modeling are developed. The following topics are covered: probability and measure, the coin-toss space and the random walk, random variables and convergence, Gaussian distribution, martingales, Brownian motion, stochastic integration for semimartingales and Ito formula, Girsanov's theorem, stochastic differential equations, continuous time market models and pricing by arbitrage, resume of Malliavin calculus, replication and pricing of contingent claims, market completeness and the fundamental theorems of asset pricing.