In the field of nanotechnology, a wide area of research is devoted to the studies about the self-assembly of semiconductor quantum nanostructures. Upon special conditions, semiconductor crystals spontaneously aggregate in nanometre sized clusters, which show unusual electronic and optical properties, thanks to their reduced dimension. Understanding the physical processes governing the formation of these structures has a major relevance in view of their application. I In this picture III-V semiconductor nanostructures have been intensively investigated because of their excellent optical quality, that allowed their use in optoelectronics. Amongst the various methods for the fabrication of nanostructures, the molecular beam epitaxy (MBE) growth is certainly one of the most important. Materials with superior quality can be grown with this technique, which also represents the best environment for the fundamental studies of surface science, due to the relative simplicity of the deposition. Our work has been dedicated to the study of the fabrication of GaAs nanostructures by the Droplet Epitaxy (DE) technique. This MBE method constitutes an alternative path for the formation of III-V nanocrystals with excellent optical properties. Indeed in contrast with the standard layer-plus-islands growth mode, this technique is based on the spontaneous formation of nanometric droplets at the substrate surface. Although this method was firstly proposed by Koguchi twenty years ago, many aspects regarding the DE fabrication remained undisclosed. In order to clarify some of these open questions, we investigated the atomic processes occurring during the formation of GaAs nanostructures by DE, being the GaAs/AlGaAs the most studied system accessible through this technique. In particular the two main steps of the growth method, the nucleation of Ga droplets and the arsenic induced transformation of droplets into GaAs crystals, have been studied, combining the MBE growth with in-situ and ex-situ ii characterization techniques. The gained knowledge permitted the full control over the size and shape of the GaAs nanostructures, which are of the greatest importance for a quantum system. Moreover the fabrication of original quantum nanostructures with complex and designable shapes was possible, by governing the physical processes occurring on the nanoscale, based on a pure bottom-up approach. This Thesis is organized as follows: Chapter 1 contains an introduction on MBE, crystal growth and GaAs material; in Chapter 2 we reviewed the main developments of DE from the original proposal to the latest results; Chapter 3 reports the experimental equipments that have been used in our work; in Chapter 4 we described the results on the formation of Ga droplets on GaAs (001) surfaces; Chapter 5 is devoted to the analysis of the processes occurring during the creation of GaAs nanocrystals and in Chapter 6 we presented the fabrication of the original structures and some of their main optical and electronic properties.