The discovery of five-fold diffraction symmetry by D. Schechtman in 1985 shocked crystallography to its roots. Incommensurable modulation of a crystal changes former identical unit cells into different ones. This radical revolution has given rise to the study of an entirely new class of matter, the quasicrystalline materials. The study of these quasi-periodic crystals brings a unique elucidation of the fundaments of crystallography and the relationship between lattice structure and macroscopic properties.This book covers the transitions between the crystalline and the quasicrystalline state. A thorough understanding of the transition from quasiperiodic to periodic lattices is essential in order to investigate the uniqueness of quasicrystals.In this well-written volume, an overview is given of the most important problems in quasicrystallography today. Leading experts provide insight into recent experimental advances achieved by studying the phase transitions between crystalline and quasicrystalline states. In the theoretical chapters one is introduced to the depth of insight into the nature of crystallography which has arisen through a rigorous understanding of the quasicrystal structure.This book is an essential source of reference for crystallographers, crystal growers and solid state physicists working in the field.