In this talk, we define derivatives of functionals on the space of continuous paths and give an introduction to path-dependent partial differential equations (PPDEs). Since the space of continuous paths is not locally compact, we cannot rely on the theory of viscosity solutions for PDEs and need to develop a new approach. We focus on the path-dependent heat equation and link it to a control problem. We will also mention new developments, challenges and applications in this field. This talk is based on joint works with Christian Keller, Nizar Touzi and Jianfeng Zhang.