Many phenomena in real life applications (i.e. physics, finance, biology) are modeled by partial differential equations (PDEs). These mathematical models are sets of 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 an overview on the construction of 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, parabolic and hyperbolic PDEs. Participation in the FoMICS block course on "Functional and Numerical Analysis" is highly recommended as preparation.