This volume contains papers presented at the Twelfth Taniguchi Symposium on the Theory of Condensed Matter, which was held at Kashikojima (in Ise­ Shima National Park), Japan, November 14-19, 1989. The general purpose of the Taniguchi Symposia is to encourage important developing, rather than established, fields in condensed matter theory. The topic of the present sym­ posium, Quasicrystais, is quite typical. In 1984, Shechtman, Blech, Gratias and Cahn discovered the icosahedral symmetry of a diffraction pattern and Levine and Steinhardt independently presented the notion of quasicrystals. Before these discoveries, Roger Penrose of Oxford University had invented a space-filling non-periodic tiling, now called Penrose tiling. These factors form a new field that had become mathematically viable by the end of 1984, and many important new ideas are still being created. In standard textbooks of solid-state science, the first chapter used to be devoted to symmetry and periodicity in crystals. Now, the textbooks should be revised; quasi-periodicity and its physical properties should be added in several chapters and almost all standard conceptions should be reconsidered. However, the facts that are known about quasiperiodicity are not enough to complete even an introductory chapter of a textbook. Revision should be extended to generalized crystallography, defects, crystal growth, electronic structure, spectral theory and localization, electron transport, spin statistics, etc. These are all topics treated in this volume.