Abstract: Nonlinear patterns and waves appear in many different systems and on vastly different scales in both time and space: despite this variability, their dynamic behavior is similar across these systems. Mathematical techniques can help identify the origins and common properties of patterns and waves across different applications. Despite many advances, understanding patterns and waves still poses significant mathematical challenges. I will show how a combination of geometric dynamical-systems ideas combined with PDE approaches can shed light on the existence, stability, and dynamical properties of nonlinear waves and will also discuss applications and open problems.