Quasicrystals are solids that exhibit symmetries long thought forbidden in nature. Since their discovery in a rapidly solidified Al-Mn alloy in 1984, the central issue in the field has been to understand why they form. Are they energetically stable compounds or stabilized by entropy? In recent years, major strides have been made in determining atomic structure, largely by direct imaging using advanced electron microscopy. One system is now known to be energetically stabilized, and quasicrystals are therefore firmly established as a new physical state of matter. They represent a unique packing of atomic clusters some tens of atoms in size, with substantial localized fluctuations, referred to as phasons. Understanding phasons may in future allow their unique macroscopic properties to be tailored for useful materials applications.