Many phenomena occurring in real life applications (i.e. physics, finance, biology) are modeled by means of partial differential equations (PDEs). These mathematical models are sets of differential equations, which describe the essential behavior of a natural or artificial system, in order to forecast and control its evolution. The aim of the course is twofold. Firstly, we will give the students an overview on the construction of differential PDEs for basic physical applications. Then, focusing on the arising PDEs, their theoretical mathematical background will be discussed. As the understanding of PDEs is closely connected to understand their physical meaning and the qualitative and quantitative behavior of their solutions, the theoretical investigations will be accompanied by the introduction of numerical schemes, which will allow for the illustrative numerical investigation of PDEs. We will consider elliptic operators (diffusion), parabolic (heat equation), and hyperbolic (fluid flow, advection).