The primary investigations of this thesis concern epitaxial studies upon quasicrystal surfaces: specifically, quasicrystal surfaces which display rotational symmetries found in periodic systems. These types of quasicrystal surfaces are often overlooked. As a result, adsorption or structural phenomena which may be unique to these systems is under-reported. This work aims to bridge this gap. Using experimental surface science techniques such as STM and LEED, adsorbates on quasicrystal surfaces with 2-fold and 3-fold rotational symmetries have been investigated. In each case, the structure of the clean substrate has also been re-evaluated either with regards to structural models or to geometric constructs such as tilings. C60 molecules deposited on the 2-fold i-Al-Pd-Mn surface were found to arrange in a formation known as a Fibonacci square grid - a quasicrystalline structure with 4-fold symmetry. This resulted in the first reported physical example of a Fibonacci square grid. A surface model was used which produces a Mn distribution which matches the geometry of the C60 network - indicating that C60 molecules adsorb exclusively at Mn atoms, motivated by electron exchange. As a consequence, an open question on the specific structure of the 2-fold i-Al-Pd-Mn surface was answered. Pb adsorption on the 2{fold and 3{fold surfaces of the i{Ag{In{Yb system was also explored. This work built on a previous study, where Pb was observed to grow in quasicrystalline layers on the 5-fold i-Ag-In-Yb surface. Each layer exhibited a unique structure, explained using planes from the bulk i-Ag-In-Yb model. Here, for the 3-fold system, Pb follows a similar adsorption scheme, where each layer is structurally similar to specific planes in the bulk. However, a difference in the density of 'available' adsorption sites in the surface plane leads to a different growth mode. Instead of a layer-by-layer mechanism, Pb was found to grow along z in a quasi-island type fashion, producing 3 dimensional nano-structures. On the 2-fold i-Ag-In-Yb surface, the density of available sites again changed the growth mode. Instead of being explained by bulk planes, Pb adopted the highly dense structure of the substrate, sitting at adsorption sites with high geometry, producing a quasi-Stranski-Krastanov type film. Additional work includes a new type of aperiodic tiling with 3-fold symmetry. The first example of its kind, this tiling is derived in a similar fashion to the Penrose tiling, and is used to re-examine the 3-fold i-Ag-In-Yb surface.