What is a "quasicrystal"? If one never came across this notion, one might be tempted to guess that this term finds its place among many other "quasi-words" created by modern physics, such as "quasiparticles", "quasimomentum", and so forth. But unlike these, which refer to "virtual" objects whose existence is merely a concept introduced to conveniently describe complex physical systems, a quasicrystal is actually something one can really touch - it is a new state of condensed matter discovered in the early eighties of the twentieth century. The "quasi" in the name refers to the fact that quasicrystals in many respects resemble conventional crystals, but differ from these in one important aspect: They are not built by a single unit cell that repeats periodically in space. It was this lack of periodicity in space, beforehand tacitly assumed for structures with a Bragg-like pure point diffraction, that delayed the acceptance of the experimental discovery and has led to many controversial discussions in the scientific community. Nevertheless, after about fifteen years, the issue appears to be settled today; and quasicrystals have become recognized as solid-state phases, in a way filling a place intermediate between amorphous systems and conventional periodic crystals, which show interesting physical properties.