With his famous discovery of quasicrystalline order in 1982–1984, Dan Shechtman, the 2011 Nobel Laureate for Chemistry, helped establish a fascinating field in materials science that has impacted also other fields in metallurgy, geology, polymer science, artificial nanostructured materials, low temperature physics, etc. The discovery forced scientists to deepen their understanding of structure and order of matter in nature and matter's properties; thereby suggesting new applications. In this article, the most salient features of this field are presented, beginning with a view of the different ways quasiperiodicity is expressed in solids, and explaining how atomic arrangements and electron transport properties may be described in aperiodic structures. Quasicrystals, as the ultimate degree of complexity in metallic compounds, are then discussed in the context of complex metallic alloys. Finally, two examples of applications—composites and catalysts—that take advantage of the specific electron transport and electron bonding in quasicrystals are mentioned.