ABSTRACT: Test functions are an important tool used to study various equations, allowing one to make explicit estimates and computations. I will introduce them and give some examples of this for Laplace’s equation. Test functions are also used to define weak solutions, which can be defined for every PDE (and ODE), and which in some cases provide the best framework in which to study an equation. I will illustrate this with a nonlinear first order PDE which models the formation and propagation of shock waves. --- ALSO: Dr. Kennedy will be leaving to take up a research position in Portugal. We will be taking him out to farewell drinks after the workshop. All welcome.