A theoretical description of nuclear spin-echo Fourier-transform mapping spectroscopy (NSEFTMS) for broad NQR lines is derived from the time-domain spin-echo theory for a spin I = 32 system. The expression found for the effective broadband RF excitation profile is similar to that in NMR spectroscopy. The main difference is the angular dependence of the nutation frequency, but this does not change the overall behavior. It is also shown that the maximum of the spin echo occurs at time t = tau + 2 2pitw1 after the second pulse, and that if the signal is acquired from this point, the phase-adjustment problem due to the off-resonance effect is avoided. A study of 35Cl NQR broad lines in molecular alloys of (p-dichlorobenzene)(1-x) (p-dibromobenzene)x at different temperatures and in the incommensurate phase of bis-(4-chorophenyl) sulfone at 77 K has been used as an example of the application of the method. The NSEFTMS method in NQR provides an efficient alternative to the time-consuming point-by-point scanning technique for the study of broad lines in solids, in particular when some sharp features exist. Copyright 1998 Academic Press.