Metal‐based icosahedral quasicrystals with perfect quasiperiodic order are found to be either semimetals or insulators. Moreover, the rapid variation of their transport properties with temperature and alloy chemistry is anomalous in comparison with disordered metals in the weakly localized regime. These results are ascribed to the collaborative effects of the band structure and strong atomic potential scattering. The band structure effect is caused by the Fermi‐surface‐quasi‐Jones‐zone boundaries interaction which is enhanced by the global icosahedral symmetry in i‐crystals,, resulting in the formation of a pseudogap. Further understanding of the i‐crystals has been advanced through studies of crystal‐analogs known as approximants. Transport studies have provided information on the conductivity spectrum &sgr;(E) near the pseudogap minimum. The pseudogap in &sgr;(E) is found to be at least an order of magnitude smaller than the pseudogap in the density of states N(E); and if &sgr;(E) varies as the &agr;th power of N(E), &agr;≳3. However, a semiconducting gap is not observed. In contrast, a real gap exists in the metal‐based crystalline compound Al2Ru. These results are attributed to the Anderson localization of electrons in a quasiperiodic lattice. For comparison, magnetoresistance data on decagonal quasicrystals are presented.