Zinc oxide (ZnO) nanostructures exhibit unique optical properties which makes them potential candidates for optoelectronic device applications. It is very important for various applications to be able to control morphology of low dimensionality nanostructures. In the first part of this thesis, synthesis and characterization of well aligned ZnO nanorods and nanorods/nanowalls nanostructures and the understanding of growth morphology are described. By using a simple vapour phase transport (VPT) and condensation process, we have grown low dimensionality ZnO nanostructures on a-plane sapphire with gold (Au) catalyst. The catalytic growth of various nanostructures is controlled by the conventional vapour liquid solid (VLS) mechanism. We demonstrate that morphology of ZnO nanostructure could be controlled by varying the parameters viz., substrates, growth duration, growth temperature, and Au catalyst thickness. Temperature dependent photoluminescence analysis indicates the ZnO nanostructures are of excellent optical quality. In the case of samples showing both nanorod and nanowall morphologies, we observe an intense excitonic emission at 3.37 eV at low temperature and the intensity decays rapidly with increasing the temperature. We propose this peculiar behaviour is due to high surface to volume ratio .associated with the nanorod/nanowall systems. The use of such nanostructures in room temperature optoelectronic devices appears to be dependent on the control or elimination of such surface effects. Arising from our studies of ZnO growth on sapphire we have also grown epitaxially ordered zinc aluminate (ZnAlz04) with sub-micron dimensions on bare c-sapphire substrates. The epitaxially oriented deposit displays the form of characteristically twinned grains. The structure of these grains and their epitaxial relation to the substrate is studied using TEM, in cross-section and plan-view, in addition to FESEM and other measurements. The growth of these grains is strongly associated with the presence of extended defects in the sapphire substrate, and can be seeded by local mechanical damage to the substrate.