A critical analysis of recent advances in synthesis and electrical characterization of nanofibers and nanotubes made of different conjugated polymers is presented. The applicability of various theoretical models is considered in order to explain results on transport in conducting polymer nanofibers and nanotubes. The relationship between these results and the one-dimensional (1D) nature of the conjugated polymers is discussed in light of theories for tunneling in 1D conductors (e.g. Luttinger liquid, Wigner crystal). The prospects for nanoelectronic applications of polymer fibers and tubes as wires, nanoscale field-effect transistors (nanoFETs), and in other applications are analyzed.