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Quasicrystals have intrigued and stimulated research in a large number ofdisciplines. Mathematicians, physicists, chemists, metallurgists and materialsscientists have found in them a fertile ground for new insights anddiscoveries. In the quarter century that has ensued since the publication ofthe experimental observation of a quasiperiodic Al-Mn alloy \cite{shecht}, manydifferent kinds of quasiperiodic alloys have been manufactured and studied. Thephysical properties of quasicrystals are no less interesting than the unusualstructural properties that led to their discovery in 1984. In this review, Ipresent some of the properties that characterize quasicrystals, briefly discussseveral types of theories that have been put forward, and describe some newbehaviors that might be investigated by experiment.

Open theoretical problems in the physics of aperiodic systems