The use of one-dimensional inverse quantum scattering theory for the design of semiconductor heterostructures is reviewed within the contexts of (i) the single-band constant effective mass model (ii) the single-band variable effective mass model and (iii) the multi-band pseudopotential model. As a fourth application a different inverse method is developed for the multi-band k · p model. This method is based on the one-dimensional coupled-channel Marchenko method. Preliminary numerical results are presented for a two-band k · p model of an InAs / GaSb / InAs single quantum well in which the strength of the interband coupling is approximately constant.