Motivated by precipitation patterns such as Liesegang rings, we discuss patterns formed in the wake of moving fronts. It turns out that strikingly simple kinetic mechanisms can lead to a plethora of patterns, from simple bands and stripes to spirals and helices. We'll discuss a number of mathematical problems that arise when one tries to predict which wavenumbers and what type of pattern would arise in the wake of fronts. Applications include elementary chemistry, patterns in bacteria colonies, and self-assembly of nano-scale textures.