Materials displaying five-fold (icosahedral) symmetries were once thought to be contrary to the rules of conventional crystallography. In this talk I describe the background to the discovery in of quasicrystals in 1984 and the discuss how their 3-d structure (as yet unknown in detail) must relate to Penrose tilings and the Fibonacci pentagrid. Quasicrystals have unusual tribological properties such as exceptional wear resistance and low coefficients of friction. The question of whether these properties stem from the quasicrystalline nature of the surface or from other causes provides the motivation for surface studies. Such studies have only recently become possible due to the development of the ability to grow large single quasicrystalline grains. I will present the results of our investigations of the clean surface structure of the AlPdMn quasicrystal and address the question of whether it is possible to form ordered 2-dimensional quasicrystalline adsorbate overlayers.