This dissertation involves the study of epitaxial behavior of one-dimensional nanomaterials like single-walled carbon nanotubes and Indium Arsenide nanowires grown on metallic catalyst surfaces. It has been previously observed in our novel microplasma based CVD growth of SWCNTs on Ni-Fe bimetallic nanoparticles that changes in the metal catalyst composition was accompanied by variations in the average metal-metal bond lengths of the nanoparticle and that in turn, affected nanotube chirality distributions. In this dissertation, we have developed a very simplistic model of the metal catalyst in order to explain the nanotube growth of specific nanotube chiralities on various Ni-Fe catalyst surfaces. The metal catalyst model is a two-dimensional flat surface with varying metal-metal bond lengths and comprising of constituent metal atoms. The effect of the composition change was modeled as a change in the bond length of the model catalyst surface and density functional theory based calculations were used to study specific nanotube caps. Our results indicated that nanotube caps like (8,4) and (6,5) show enhanced binding with increased metal-metal bond lengths in the nanoparticle in excellent agreement with the experimental observations. Later, we used this epitaxial nucleation model and combined with a previously proposed chirality-dependent growth rate model to explore better catalysts that will preferentially grow an enhanced chirality distribution of metallic nanotubes. From our DFT calculations and other geometrical considerations for nanotube growth, we demonstrated that the pure Ni0.5Cu0.5 metal nanoparticles and its lattice-strained surfaces can serve as a promising catalyst for enhanced growth of metallic nanotubes. Finally, we extended this model of epitaxial growth to study the growth of,andoriented nanowires on gold metal nanoparticles where a faster growth rate ofnanowires was previously observed in experiments on shaped nanoparticles than that on spherical nanoparticles. The DFT calculations indicated an enhanced growth selectivity of theoriented nanowires on the Au(111) surfaces. However, the DFT results also show that theandNWs will preferentially grow on the Au(100) surface than on the Au(100) surface. The epitaxial model based DFT calculations of nanotube and nanowire growth on metal catalyst surfaces presented in this dissertation, provide a deep insight into their epitaxial growth mechansims and, can be easily exploited to layout better design principles of synthesizing catalysts that helps in growing these one-dimensional nanomaterials with desired material properties.