The formation of nanostructures during metalorganic vapor-phase epitaxy on patterned (001)/(111) B GaAs substrates is reviewed. The focus of this review is on the seminal experiments that revealed the key kinetic processes during nanostructure formation and the theory and modelling that explained the phenomenology in successively greater detail. Experiments have demonstrated that V-groove quantum wires and pyramidal quantum dots result from self-limiting concentration profiles that develop at the bottom of V-grooves and inverted pyramids, respectively. In the 1950s, long before the practical importance of patterned substrates became evident, the mechanisms of capillarity during the equilibration of non-planar surfaces were identified and characterized. This was followed, from the late 1980s, by the identification of growth rate anisotropies (i.e. differential growth rates of crystallographic facets) and precursor decomposition anisotropies, with parallel developments in the fabrication of V-groove quantum wires and pyramidal quantum dots. The modelling of these growth processes began at the scale of facets and culminated in systems of coupled reaction-diffusion equations, one for each crystallographic facet that defines the pattern, which takes account of the decomposition and surface diffusion kinetics of the group-III precursors and the subsequent surface diffusion and incorporation of the group-III atoms released by these precursors. Solutions of the equations with optimized parameters produced concentration profiles that provided a quantitative interpretation of the time-, temperature-, and alloy-concentration-dependence of the self-ordering process seen in experiments.