The goal of this chapter is to develop the mathematical theory of time-dependent equations as preparation for PDE models in computational finance. We focus on the two-dimensional heat equation because first, its study follows naturally from previous chapters that discuss the Laplace equation and its associated properties and boundary value problems. Second, we lay the foundations for work on convectiondiffusion-reaction equations and their numerical approximation by the finite difference method. In particular, the maximum-minimum principle for PDEs and finite difference schemes will be important. We also discuss the concept of well-posed problems.