Quasicrystals are metallic alloys that possess perfect long-range structural order, in spite of the fact that their rotational symmetries are incompatible with long-range periodicity. The exotic structural properties of this class of materials1 are accompanied by physical properties that are unexpected for metallic alloys. Considerable progress in resolving the geometric structures of quasicrystals has been made using X-ray and neutron diffraction, and concepts such as the quasi-unit-cell model2 have provided theoretical insights. But the basic properties of the valence electronic states—whether they are extended as in periodic crystals or localized as in amorphous materials—are still largely unresolved3. Here we investigate the electronic bandstructure of quasicrystals through angle-resolved photoemission experiments on decagonal Al71.8Ni14.8Co13.4. We find that the s-p and d states exhibit band-like behaviour with the symmetry of the quasiperiodic lattice, and that the Fermi level is crossed by dispersing d-bands. The observation of free-electron-like bands, distributed in momentum space according to the surface diffraction pattern, suggests that the electronic states are not dominated by localization.