In this chapter, we will examine how diffusion can interact with the sort of nonlinear reaction kinetics that we have discussed in previous chapters to generate propagating waves of chemical reactivity and structured, sometimes spectacular, spatial patterns. One might argue, particularly if one were a physicist rather than a chemist, that the oscillatory phenomena that we have discussed thus far, and even the chaotic behavior that we will treat a bit later, can be just as easily generated by nonlinear electrical circuits, and that those circuits are much easier to work with than wet, smelly, and often toxic, chemicals. However, as far as chemistry goes, “spatial is special” in the sense that the spatial phenomena that we will discuss in this chapter are most naturally and conveniently generated in chemical systems. These same patterns also occur in a number of biological systems; the resemblance between a structure that develops in an inorganic chemical reaction and one found in a living organism can be startling. We are inclined to think of diffusion as tending to make a system more homogeneous, with it acting to reduce and ultimately to obliterate local fluctuations in the concentration of a chemical species. The spontaneous appearance of concentration gradients might seem to violate the Great Law (i.e., the Second Law of Thermodynamics), since the free energy should be at a minimum or the entropy at a maximum when concentrations are uniform. However, as we have seen in the case of temporal oscillation, a system far from equilibrium can generate spontaneous transient spatial inhomogeneities and, in an open system, sustained spatial patterns as well. The most common kind of chemical wave is the single propagating front, where, in an unstirred medium, there is a relatively sharp boundary between reacted and unreacted material, and this boundary or wavefront moves through the solution at an essentially constant speed. These waves are, of course, most noticeable when the reaction involves a color change. Many autocatalytic reactions exhibit this phenomenon of traveling waves.