Abstract <p>The fluctuation acceleration in the lunar dusty plasma has been considered in a situation where the effects of solar radiation are not important, which occurs, e.g., during the dark time of the lunar day. Acceleration is caused by fluctuations in the charges of dust particles, which arise as a result of the discrete nature of the dust causing fluctuations in the microscopic currents of electrons and ions flowing onto the dust particles. Fluctuations in the charges of dust particles lead to an increase in the average energy of the particles and, accordingly, to their acceleration. The energy that makes possible such an increase in the energy of dust particles is associated with sources of dusty plasma, which ensure a continuous exchange of matter and energy including the participation of dust charging processes. It has been shown that in the lunar dusty plasma, fluctuation acceleration of nanoscale and submicrometer dust particles can lead to a significant increase in their speed, and for sufficiently small nanoscale particles, even to their reaching speeds exceeding the second astronautic velocity for the Moon, as a result of which such particles can leave the lunar plasma-dust system. At the same time, earlier estimates characterizing the behavior of the largest particles, which make up the dusty plasma formed by electrostatic interactions, remain valid.</p>