In nonlinear systems a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wave vectors and the dispersion equation are met. The simplicity of the criteria make technique popular in multiple fields. Its most prominent and well-developed forms appear in the study of gravity waves, but also finds numerous applications from astrophysics and biology to engineering and medicine. Theoretical work on partial differential equations provides insights into chaos theory; there are curious links to number theory. Resonant interactions allow waves to (elastically) scatter, diffuse or to become unstable.[1] Diffusion processes are responsible for the eventual thermalization of most nonlinear systems; instabilities offer insight into high-dimensional chaos and turbulence.