CC BY-SA 4.0 UnportedBramburger, Jason J.2024-10-162024-10-162023https://oa.tib.eu/renate/handle/123456789/16870https://doi.org/10.34657/15892In this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social processes. We begin by focussing on two types of patterns that do not change in time: space-filling patterns and localized patterns. We then discuss two types of waves that evolve predictably as time goes on: spreading waves and rotating waves. All our examples are motivated with real-world applications and we highlight some of the main lines of research that mathematicians pursue to better understand them.enghttp://creativecommons.org/licenses/by-sa/4.0/510AnalysisNumerics and Scientific ComputingReport