With reduced size and dimensions, semiconductor nanostructures have dramatic difference in electrical properties than their bulk counterparts. These differences have raised many interesting topics, such as quantum Hall effects and bandgap engineering. By utilizing these properties of nanostructures, numerous electrical devices have been proposed and created to improve human life quality and production efficiency. Graphene, carbon atoms formed in only one-atomic-layer, has been discovered in lab in 2004 and raised a lot of attentions. It has unique electronic band structure and promising applications. Many fundamental physics questions and practical limitations on device functions need to be addressed. We have investigated the low-frequency 1/f noise, which poses a limit on the signal-to-noise ratio in broad band electrical circuit, in both suspended and on-substrate graphene field-effect transistors, from T = 30 to 300K. We have found that, compared to on-substrate devices, in general suspended graphene devices show lower 1/f noise, as a result of their higher mobility. We explain the observed noise dependence on gate voltage using the Hooge's empirical relation with a variable Hooge parameter. We have also studied metal-insulator transition in monolayer graphene, and a phase diagram of metal and insulator phases from reported and our own results was proposed. Anomalous quantum Hall effect in ABC-stacked trilayer graphene was also studied, and we found chiral fermions with Berry's phase 3 pi and unusual Landau level quantization with 12-fold and 4-fold degeneracy in the n=0 and n >0 Landau levels, respectively. With the help of spacially resolved scanning photocurrent, we have studied charge transport in graphene in quantum Hall regime. And we found that the net photocurrent is determined by hot carriers transported to the device edges, where quantum Hall edge states provide efficient channels for their extraction to contacts. Moreover, optical bandgap and electrical properties of (GaN)1-x(ZnO)x solid solution nanowire have been studied to explore the possibility of using this material as a photovoltaic catalyst in water splitting for generating renewable energy. The nanowires with composition x = 0.12 were found with an optical bandgap ~2.7eV. And the nanowire FETs were n-type conduction, with background carrier density ~10^19/cm^3 and electron mobility ~1 cm^2/V-s.