These 28 contributions by leading researchers - from such diverse disciplines as chemistry, biology, physics, mathematics, and physiology - describe recent experiments, numerical simulations, and theoretical analyses of the formation of spatial patterns in chemical and biological systems.Chemical patterns have been systematically studied since the field was established by Alan Turing's landmark 1952 paper, "The chemical basis for morphogenesis," yet only recently have new experimental techniques and numerical analyses of reaction-diffusion equations opened the way to understanding stationary and traveling wave patterns.This collection summarizes the exciting developments in this rapidly growing field. It shows that some biological patterns have been found to be strikingly similar to patterns found in simple, well-controlled laboratory chemical systems, that new chemical reactor designs make it possible to sustain chemical patterns and to study transitions between different kinds of patterns, and that nearly 40 years after Turing's paper, the patterns predicted by Turing have finally been observed in laboratory experiments.Harry L. Swinney is Sid Richardson Foundation Regents Chair, Department of Physics, and Director of the Center for Nonlinear Dynamics at the University of Texas at Austin. Valentin I. Krinsky is Head of the Autowave Laboratory, Institute of Biological Physics, Academy of Sciences, Pushchino, USSR.Chapters cover: Spiral, Ring, and Scroll Patterns: Experiments. Spiral, Ring, and Scroll Patterns: Theory and Simulations. Fronts and Turing Patterns. Waves and Patterns in Biological Systems.