Although the main international research thrust on self-assembled epitaxial semiconductor quantum dots is currently being directed towards random alloy quantum dots, the suggestion is made that atomically ordered quantum dots which are grown by either epitaxy or endotaxy may in addition to their larger quantum confinement potentials possess superior long term structural stability. Such atomically ordered quantum dots should, therefore, be superior to random alloy quantum dots as far as prospective device applications are concerned. The basis for this suggestion is simple thermodynamic considerations. These considerations seem to explain our transmission electron microscopical observations of epitaxially grown atomically ordered In(Sb,As), (In,Ga)Sb, (Cd,Zn)Se, (Cd,Mn,Zn)Se quantum dots and Pb(Se,Te) quantum dot predecessor islands. Atomic ordering in (In,Ga)P quantum dot structures, as recently observed by other authors, does not seem to contradict our thermodynamic considerations. Endotaxially grown atomically ordered (In,Si,As) and (Sn,Si) quantum dots in Si matrices are briefly discussed as an even more unconventional approach to nanostructures with applications in electronics, photonics, information storage, and sensing.