In recent years the vapour phase growth of quasi one-dimensional (quasi-1D) semiconductor nanocrystals (nanowires and related quasi-1D nanostructures) has gathered considerable interest due to their importance for both fundamental physical studies and potential applications to nanoscale electronics and photonics. In particular, the fabrication of free-standing nanowires epitaxially oriented on a crystalline substrate is in the focus of most research efforts throughout the world, as these structures allow minimal interaction with the underlying substrate and allow to evidence novel 1D quantum size effects. A most successful approach to the bottom-up fabrication of these quasi-1D nanostructures is the use of self-assembly growth methods, based on the so-called metal catalyst assisted Vapour-Liquid-Solid (VLS) mechanism. In this paper, we briefly review the different approaches to the VLS self-assembly of quasi-1D nanostructures of compound semiconductors based on vapour phase epitaxy (VPE) technologies. The current state-of-the-art in the field is discussed for nanowires of both III-V compounds and ZnO; in particular, results obtained in the author’s own laboratory on the VLS self-assembly process of epitaxial GaAs nanowires using metalorganic VPE, and ZnO nanowires by solid-source VPE are presented.