The state-of-the-art of the theory of electronic properties of quasicrystals is reviewed. Although the existing calculations are generally in very good agreement with experimental spectroscopic data, open questions concerning the character (extended, critical, localized) of the eigenstates in real three-dimensional quasicrystals remain. As long as this fundamental problem subsists, a deeper understanding of certain aspects of electronic transport, magnetism, etc. will remain incomplete.
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