  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.
