We review different strategies to achieve a three-dimensional energy bandgap modulation in a nanowire (NW) by the introduction of self-assembled 0D, 1D and 2D quantum structures, quantum dots (QDs), quantum wires (QWRs) and quantum wells (QWs). Starting with the well-known axial, radial (coaxial/prismatic) or polytypic quantum wells in GaN/AlN, GaAs/AlAs or wurtzite/zinc-blende systems, respectively, we move to more sophisticated structures by lowering their dimensionality. New recent approaches developed for the self-assembly of GaN quantum wires and InAs or AlGaAs quantum dots on single nanowire templates are reported and discussed. Aberration corrected scanning transmission electron microcopy is presented as a powerful tool to determine the structure and morphology at the atomic scale allowing for the creation of 3D atomic models that can help us to understand the enhanced optical properties of these advanced quantum structures.