A theory of photo- and dark-band conductivities in semiconductor supercrystals consisting of nanocrystals is developed by assuming scattering by structural defects in the supercrystals. A new proposed mechanism of photoexcitation, which is triggered by an efficient Auger ionization of charged nanocrystals, provides explanation for the measured photocurrent being 2-3 orders of magnitude larger than the dark current. For dark conductivity, the metal-insulator transitions and temperature dependence of mobility in the metal phase are considered.