Quasicrystals are aperiodically ordered solids. Their unusual transport properties, especially the low electric conductivity, are indications of the particular electronic structure of these materials. There are several approaches toward understanding the electronic structure of these systems. Whereas a complete understanding is still lacking, quasicrystals can heuristically be described as Hume-Rothery alloys, which qualitatively account for the experimentally observed properties. Other approaches that have been followed are based on the investigation of electrons in aperiodic potentials and on a hierarchical cluster picture of quasicrystals.