Quasicrystals are ordered materials which present a long range orientational order but no translational order. Their electronic properties are spectacular. For instance, the AlPdRe phase exhibits a semiconductor like resistivity (> 10 Ohm.cm at 4 K). However, the density of states at the Fermi level is around one tenth of the one of pure aluminum. Ab-initio calculations of the electronic structure in several inter-metallic aluminum based alloys and approximants of Quasicrystals are presented. The calculated densities of states exhibit the following characteristics: a well pronounced pseudogap at the Fermi level (as expected for Hume-Rothery Alloys); a spiky structure. The consequences of the spiky structure on transport properties are analyzed. The role of transition metals is studied by a generalized virtual bound state model taking the structural properties of the alloys into account. We present an explanation of the effective negative valence which is usually mentioned in inter-metallic alloys. Moreover, beside the Hume-Rothery Alloy effect, we show that the transition metal atoms help in reducing the density of states at the Fermi level. Within the multiple scattering theory, the role of the local atomic order on the localization of the electrons in Quasicrystals is modelized considering atomic cluster embedded in a metallic matrix. This model leads to "cluster virtual bound states". This is consistent with the spiky structure of the density of states determined from ab-initio technique and the notion of critical states which is characteristic of the electronic spectrum of quasiperiodic tilings. However, we show that the cluster stability which mainly results from pair interactions is not due to localization. This point is consistent with the usual picture for which Quasicrystals are Hume-Rothery alloys.
