It is shown that multiple-quantum NMR spectroscopy for quadrupolar spins in powder solids gives enhanced resolution over single-quantum spectra for the case of triple-quantum (TQ) NMR for spin-3/2. The problem of generating TQ coherence evenly over all molecular orientations is considered and solutions are found. The effects of sample spinning on the excitation of the TQ coherence for spin-3/2 is considered in detail. It is shown that selection of a suitable spinning speed is essential if sensible spectra are to be recorded. Finally, it is predicted that the effects of the quadrupole interaction can be removed completely from triple-quantum spectra of spin-3/2 by spinning the sample at an angle of 70.1 degrees or 30.6 degrees with respect to the applied field, and experimental results are given. Conclusions derived in this work are applicable to the newly termed multiple-quantum magic-angle-spinning experiment. (C) 1997 Academic Press.