Quasicrystals are metallic alloys which posess long-range order but not periodicity; hence they may display structural symmetries not observed in periodic materials. The surfaces of these materials offer opportunities for the fabrication of nanostructures and thin films which themselves have unusual symmetries and structures. In turn, measurements of such nanostructured systems may offer insights into the larger question of the relationship between physical properties and aperiodicity. Several such systems, ranging from individual atoms to nanoclusters to thin films, have been fabricated and are under investigation in our laboratory. The growth of Si on quasicrystal surfaces is one recent example. On the AlPdMn quasicrystal, at submonolayer coverages, individual Si atoms occupy a unique adsorption site; thus the Si atoms can be considered to be arranged in a two-dimensional quasiperiodic array. On decagonal AlNiCo, the Si atoms form 5-fold nanoclusters, again indicative of a preferred adsorption site. Another spectacular example is that of an ultrathin film grown by the deposition of copper atoms on the five-fold surface of the icosahedral Al@sub 70@Pd@sub 21@Mn@sub 9@ quasicrystal.@footnote 1@ STM images show that the in-plane structure comprises rows having separations of S=4.5±0.2 Ã. and L=7.3±0.3 Ã., whose ratio is the golden mean @tau@=1.618... within experimental error. The sequences of such row separations form segments of terms of the Fibonacci sequence, indicative of the formation of a pseudomorphic Cu film. Finally, we have recently demonstrated that pseudomorphic films can also be grown using magnetic elements such as Co, Fe and Ni. Atoms in a quasiperiodic pattern are predicted to form new types of frustrated structures; such films therefore offer the possibility of observing these novel magnetic effects. @FootnoteText@@footnote 1@J. Ledieu, J.T. Hoeft, D.E. Reid, J.A. Smerdon, R.D. Diehl, T.A. Lograsso, A.R. Ross and R. McGrath, Phys. Rev. Lett. 92 (2004) 135507.