The aim of this thesis is to understand the dynamics of the growth of catalyst-free III-V semiconductor nanostructures on silicon substrates, focusing the attention mainly on the early stages of growths. A detailed understanding of this first phase of the process is a key step to obtain a completely successful integration of highly functional materials, like the III-Vs, on the CMOS platform, and to synthesize nanostructures with tailored properties. Indeed, the first mandatory step in nanotechnology is the possibility to fabricate nanostructures and nanomaterials, i.e. structures and materials with at least one dimension falling in the nanometer scale. Among different nanostructures, semiconductor nanowires (NWs) have proven to be versatile building blocks for a manifold of applications. In this work two types of nanostructures have been investigated: NWs and V-shaped nanomembranes. Their growth has been performed by molecular beam epitaxy (MBE), a technique that allows to produce ultrapure nanostructures, with very high crystalline quality and atomically sharp interfaces. The growth has been obtained with a self-catalyzed approach, meaning that no external material, apart the constituents of the semiconductor to be grown, has been used. This allows to avoid any possible contamination of the substrate, a fundamental requirement to ensure a full compatibility with the silicon technology. We performed a systematic study on the growth directions of self-catalyzed GaAs NWs grown on silicon substrates. Indeed, when growing III-V NWs on silicon non vertical wires always appear. In order to make NW-based structures on silicon effective for applications like energy harvesting (one of the most promising so far) it is mandatory to control the growth direction and in particular to maximize the yield of NWs perpendicular to the surface. By analyzing the first stages of growth we shed light on the mechanism responsible for the different NWs orientations: we optimized the growth conditions to obtain up to a 100% yield of vertical NWs. Having attained growth of nanostructures in an ordered manner along regular arrays is a subsequent step for the fabrication of devices. In this work our capability to control growth of InAs and GaAs NWs in arrays is presented and the existing challenges for the reproducible growth are highlighted. Lastly, we turned to the growth of nanostructures on exactly oriented (001) substrates, which would make the III-V/group IV integration compatible with the current technological pro- cesses. This led us to the discovery of a new class of III-V semiconductor nanostructures, called V-shaped nanomembranes, characterized by a unique morphology and growth mech- anism and possessing interesting optical properties. An accurate characterization of their morphology and a complete understanding of their nucleation and growth mechanism is reported. These results give a clear pathway of how to obtain fully controlled structures which as such could be useful for the realization of complex branched interconnected nanoelectronic devices.