The acceleration of Auger-type multicarrier recombination in semiconductor nanocrystals impedes the development of many quantum-dot photonics, solar-cell, lighting, and lasing technologies. To date, only multiexciton and charged-exciton Auger recombination channels are known to show strong size dependence in nanocrystals. Here, we report the first observation of strongly accelerated "trap-assisted" Auger recombination rates in semiconductor nanocrystals. Trap-assisted Auger recombination in ZnO nanocrystals, involving the recombination of conduction-band electrons with deeply trapped holes via nonradiative energy transfer to extra conduction-band electrons, has been probed using time-resolved photoluminescence and transient absorption spectroscopies. We demonstrate that this trap-assisted Auger recombination accelerates dramatically with decreasing nanocrystal size, having recombination times of >1 ns in the largest nanocrystals but only ~80 ps in the smallest. These trap-assisted Auger recombination rates are shown to scale with inverse nanocrystal radius squared (1/τ(Auger) ~ R(-2)). Because surface carrier traps are ubiquitous in colloidal semiconductor nanocrystals, such fast trap-assisted Auger recombination is likely more prevalent in semiconductor nanocrystal photophysics than previously recognized.