A critical analysis of recent advances in the 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 nanofibers and tubes as wires, field-effective transistors (nanoFETs), and in other applications are analysed.