This course explores advanced topics at the forefront of mathematical finance. Core themes include the theory and applications of forward-backward stochastic differential equations, stochastic optimal control, and the Hamilton-Jacobi-Bellman equation. Additional topics covered are the stochastic Pontryagin principle, finite player and mean field games, models driven by Lévy processes, and contemporary topics chosen by the instructor. Emphasis is placed on both the mathematical foundations and their relevance to modern problems in finance.