Open theoretical problems in the physics of aperiodic systems

Quasicrystals have intrigued and stimulated research in a large number of disciplines. Mathematicians, physicists, chemists, metallurgists and materials scientists have found in them a fertile ground for new insights and discoveries. In the quarter century that has ensued since the publication of the experimental observation of a quasiperiodic Al-Mn alloy \cite{shecht}, many different kinds of quasiperiodic alloys have been manufactured and studied. The physical properties of quasicrystals are no less interesting than the unusual structural properties that led to their discovery in 1984. In this review, I present some of the properties that characterize quasicrystals, briefly discuss several types of theories that have been put forward, and describe some new behaviors that might be investigated by experiment.