This paper is an attempt to present a chronological review of the structural concepts that have been developed to characterize quasicrystals, starting from the experimental discovery of D. Shechtman [1] and the concomitant theoretical definition of quasicrystal as proposed by D. Levine and P. Steinhardt [2], up to the present research in the field. The largest part of the paper is a discussion of the specific points that make the atomic structure determination of quasicrystals an original and

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... fficult scientific challenge. We finally discuss the soundness of our knowledge of the actual atomic structure in quasicrystals: a quite solid idea of where the atoms are but a not yet secure vision of the chemical species distribution.