Low dimensional semiconductor structures exhibit novel physical properties which resulted already in devices like quantum well lasers, high electron mobility transistors, solar cells, etc. With shrinking dimensions of these structures there is a need for physical characterization on the nanometer scale. At this point, modern scanning probe microscopy comes into play: The conductive probe based atomic force microscopy techniques like conductive atomic force microscopy (C-AFM), Photoconductive AFM (PC-AFM) and Kelvin probe force microscopy (KPFM) provide an opportunity to study electrical and electronic properties with nanometer resolution. In this thesis, the application of these techniques for a variety of substrate supported and upright semiconductor nanostructures like nanowires (NW), nanodots (ND), nanosized defects and free standing nanorods (NR) is demonstrated. Using C-AFM under ultra-high vacuum (UHV) conditions, we examined samples containing InAs NWs and NDs grown on GaAs(110) substrates by molecular beam epitaxy. It has been demonstrated that in the case of the conventionally grown sample, InAs decorates step bunches running along [1-10] direction. For the case of H-assisted grown sample, InAs forms NDs at the apexes bounded by steps running along [1-15] and [-115] directions. These results demonstrate that C-AFM can be used for the exploration of III-V semiconductor nanostructures beyond the limits of conventional AFM. Application of C-AFM to study so called arrowhead defects (ADs) in GaInP thin lms epitaxially grown on vicinal Ge(100) surfaces revealed that the two terminating planes of the ADs exhibit increased conductivity compared to the defect free lm. Although both terminating planes appear morphologically equivalent and are composed from f105g and f109g facets, they dier in their electrical behavior as is also conrmed by KPFM. For the rst time, C-AFM and PC-AFM have been applied to study the electrical properties of upright standing ZnO NRs. I-V characterization on a single NR utilizing C-AFM resulted in dierent Schottky barrier heights for top and side planes. The obtained values agree with macroscopic data known from the literature. The PC-AFM measurements performed on single ZnO NRs exhibit the persistence of photocurrent with large decay times. The photocurrent spectroscopy performed on single NRs reveals defect mechanism of photoconductivity. The experimental ndings stay in a good agreement with theoretical predictions from literature on the role of oxygen vacancies in the persistent photoconductivity of ZnO.