The study of nonlinear wave propagation and pattern formation in natural systems is a prime example of the insight that mathematics can offer into the qualitative and quantitative features of emergent phenomena in complex systems. As these phenomena are often inherently nonlinear, the complexity of the underlying system can prohibit mathematical analysis. However, when there is a clear spatial and/or temporal scale separation present in the system under consideration, the toolbox of geometric singular perturbation theory can be used to gain additional insight into the structure and dynamics of waves and patterns.

I will give an overview of the main ideas and techniques of geometric singular perturbation theory, and demonstrate how these can be used to construct several types of patterns in general systems of reaction-diffusion equations. Moreover, I will explain how the same techniques can give insight into the dynamics of such patterns. In addition, I will discuss a number of recent applications in several biological contexts.