Abstract Some recent results concerning domain structure and domain-wall motion in amorphous soft ferromagnets are reviewed. The effect of the value of the saturation magnetostriction is discussed, with particular emphasis on the change in sign of λs induced in nearly-zero magnetostriction alloys by large tensile stresses. The ‘fast’ permeability relaxation observed during high-frequency domain-wall motion is explained in terms of a domain-wall instability under extreme velocity conditions. Finally, the unique kinetic features of the slow relaxation (after-effect) of the initial magnetic permeability will be outlined. A specific experiment, conceived to check the assumptions of the model established for this relaxation process, is proposed and discussed in detail.