The goal of the talk is to give a broad overview of Interpolation theory, which has a rich history here in Jena. Interpolation theory, and more generally the study of function spaces, is a beautiful subject on its own, but also has proven to be extremely useful in applications such as stochastics, numerics, and (S)PDEs. In this talk, we will also see how interpolation theory can be used to study parabolic PDEs, in a modern vector-valued approach.