This review volume provides the most up-to-date and authoritative description of research on icosahedral solids, which has advanced rapidly since the discovery of these unique materials in 1984. The present book, intended as a companion volume to the reprint volume on The Physics of Quasicrystals edited by P Steinhardt and S Ostlund, will be invaluable to graduate students and workers in the field as a comprehensive reference. Scientists in related fields can use it as a readable introduction to the important current problems in quasicrystals. The chapters have been written by many of the most prominent theorists and experimentalists on quasicrystals, both physicists and materials scientists, from around the world. Especially exciting are the details of the recent discovery of "perfect quasi-crystals", new materials which promise to be an ideal form of quasiperiodic matter with little or no disorder. Other topics include: electron, X-ray and neutron quasi-crystallography, scanning tunneling microscopy studies, electronic transport experiments, quasicrystal faceting and statistical mechanics, growth rules and matching rules for quasicrystals, group theory and elasticity theory.