This thesis reports on the chemical vapour deposition (CVD) growth of carbon nanostructures (mainly carbon nanotubes (CNTs)) on perovskite oxide surfaces with the aid of various catalysts. Two types of perovskite oxide, single crystal SrTiO3 (001) and polycrystalline BaSrTiO₃, have been used as catalyst supports (in metal-catalyst-involved CVD routes) or as catalysts (via metal-catalyst-free CVD routes) for the growth of carbon nanostructures. In metal-catalyst-involved cases, SrTiO₃ (001) single crystal has been proven, for the first time, to serve as a substrate for the growth of CNTs. Fe and Ni catalysts can be tailored in a controllable manner on SrTiO3 (001) surfaces prior to the CNT synthesis, forming truncated pyramid shaped nanocrystals with uniform size distributions. The growth of vertically aligned CNT carpets was realised with the aid of Fe on SrTiO₃ (001) surfaces, and it was further found that the CNTs grow via a base growth model. Furthermore, it is possible to grow helical carbon nanostructures on BaSrTiO3 substrates by introducing a Sn catalyst into the system. The synthesised helical carbon nanostructures follow a tip growth mode, where the structural and chemical aspects of catalyst particles gave rise to a wide range of carbon morphologies. CNTs were also grown on single crystal SrTiO₃ (001) and polycrystalline BaSrTiO3 substrates via metal-catalyst-free routes. The surface-roughness-tailored growth of CNTs was surprisingly achieved on a series of engineered SrTiO₃ (001) surfaces, where a correlation between the surface roughness/morphology of the substrates and the relevant catalytic activity was revealed. The growth of CNTs arises because the catalyst fabrication methods lead to the formation of SrTiO₃ asperities with nanoscale curvatures, over which the CNTs are generated throughout a lift-off process. Facet-selective growth of CNTs was observed on polycrystalline BaSrTiO₃ surfaces, where BaSrTiO₃ (110) facets lead to the growth of CNTs on them, whereas the (001) facets result in no growth at all. This observation was further analysed in the content of the adsorption and diffusion of carbon species on distinct BaSrTiO₃ facets, before reaching the conclusion that the formation of CNTs occurs through a metal-free, stack-up process driven by the assembly of the carbon fragments.