This course provides a rigorous introduction to the modern theory of mathematical finance in incomplete markets. The first part of the course covers the fundamental theorem of asset pricing, as well as super-hedging of contingent claims, in full generality for discrete time models. The second half of the class focuses on optimal investment, pricing and hedging in continuous time incomplete markets. Here, lectures will cover optimal investment and duality for general semi-martingale models; and portfolio optimization and pricing in Markovian factor models, non-Markovian Brownian models, models with trading constraints, and models with transactions costs. Throughout this half, lectures will highlight connections to the dynamic programming principle, the Martingale optimality principle, semi-linear Cauchy partial differential equations, backward stochastic differential equations, and the theory of viscosity solutions.