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Steady-state currents flowing through planar polymer layers under irradiation with 15–50 keV electrons were studied experimentally and theoretically. The ultimate range of electrons was somewhat below the layer thickness. The Monte Carlo method was used to determine the basic transport characteristics of fast electrons in polymers (maximum range, depth distribution of absorbed dose and forward current). It was shown that significant steady-state currents (1 to 10% of the electron beam current) were observed only if the thickness of blocking (unirradiated) layer did not exceed 5 μm. The magnitude of these currents was almost unaffected by the polymer type (polymers with minimum radiation-induced conductivity and polymers with electron or hole conductivity were examined). It was found that conventional theories of conductivity of dielectrics failed to explain the observed experimental data. Additional arguments in favor of the hypothesis of “streamer” mechanism of injection currents through an unirradiated polymer layer were obtained. It is emphasized that the radiation-induced heating of polymer samples can play an important role in the phenomenon under study, acting as an undesirable technical factor, that strongly distorts obtainable experimental data.