In this thesis, the electronic structures of low-dimensional carbon allotropes have been studied. In particular, the spatially-resolved photocurrent responses of devices comprising carbon nanostructures were investigated through scanning photocurrent microscopy (SPCM). Experiments were first performed on individual semiconducting single-walled carbon nanotubes, which were simultaneously used to explore the scope of this technique. The effects of the drain-source and gate voltages on the photocurrent response of the nanotubes were systematically studied, and the resulting images revealed that the photoresponses reflect the distribution of the electrostatic potential within the tubes, thus demonstrating the potential of SPCM as a tool for the determination of the electronic band structure profile of such nanodevices. In addition, it was verified that the gate dependent evolution of the nanotube bands agree remarkably well with the widely accepted Schottky barrier transistor model for nanotube-based field-effect transistor devices. Moreover, evidence for the p-type doping of the nanotubes under ambient conditions was obtained. The photoresponse signal could also be employed for the estimation of the Schottky barrier height at the metal contacts. SPCM measurements on carbon nanotube networks revealed a highly localized photoconductive response at a few of the constituting crossed nanotube junctions, thus evidencing that the electrostatic potential drops in a rather inhomogeneous manner within the networks. In order to shine light on the origin of this localization and its implication for the electrical transport in these technologically important devices, individual crossed junction devices were extensively investigated. Zero drain-source bias photocurrent images enabled the direct observation of Schottky barriers and isotype p-p heterojunctions at metallic-semiconducting and semiconducting-semiconducting nanotube crossings, respectively. Furthermore, electrostatic potential profiles obtained from gate dependent SPCM images showed that the metal contacts dominate the dark- and photo-current response in the ON states, while the inter-tube crossings play a more important role in the OFF state. This conclusion provides valuable insights into the electrical response of the corresponding crossed junction devices. Finally, SPCM was successfully applied to evaluate the impact of the electrical contacts and the sheet edges on the properties of graphene devices. In analogy to the case of carbon nanotubes, strong photocurrent responses were detected around the contacts, thus evidencing the presence of metal-induced doping of the graphene flake. By analyzing the intensity of the photoresponses as a function of gate voltage, the Fermi level shift induced by the graphene doping could be estimated. Moreover, gate dependent photocurrent images revealed that the n- to p-type transition does not occur homogeneously within the graphene sheet. Instead, an n-type channel located at the center of the graphene flake, surrounded by p-type conducting edges, was observed in the vicinity of the Dirac point. The invasive nature of the metal contacts on graphene was further revealed by SPCM experiments on multi-terminal devices.