This course begins with a brief introduction to classical calculus of variations and its applications to classical problems such as geodesic trajectories and the brachistochrone problem. Then, we examine Hamilton-Jacobi equations, the role of convexity and the classical verification theorem. Next, we illustrate the lack of classical solutions and motivate the definition of viscosity solutions. The course ends with a brief description of the reinforcement learning problem and its connection with Hamilton-Jacobi equations.