to Professor Dr. Roland Zimmermann on the occasion of his 60th birthday Inhomogeneous broadening is a long standing issue in the physics of nanostructures, in that it prevents the detailed assessment of their electronic and optical properties, and limits the performances of nanostructure based devices. Advances in scanning probe spectroscopies have recently permitted to overcome this problem, thus opening the way to a detailed understanding of the single nanostructure properties, by virtue of their unsurpassed capability to inject carriers or photons at local scale. In particular, luminescence spectroscopy and current-voltage spectroscopy following highly localized carrier injection by a scanning tunneling microscope into a single quantum dot (QD), are the methods of choice to overcome the inhomogeneous broadening effects due to size and compositional non-uniformity. In this way one can map the eigenstates of a single dot, elucidate the relationships between carrier injection, capture and recombination, investigate the dynamics of single-dot filling, the formation of excitons and charged excitons, the occurrence of dot-dot interactions leading to molecular states, and the diffusion of charges into neighboring dots in a grand ensemble.