Why are quasicrystals quasiperiodic?

In the past two decades significant progress has been made in the search for stable quasicrystals, the determination of their structures and the understanding of their physical properties. Now, quasiperiodic ordering states are not only known for intermetallic compounds, but also for mesoscopic systems such as ABC-star terpolymers, liquid crystals or different kinds of colloids. However, in spite of all these achievements fundamental questions concerning quasicrystal formation, growth and stability are still not fully answered. This tutorial review introduces the research in these fields and addresses some of the open questions concerning the origin of quasiperiodicity.