An overview of the cleaved edge overgrowth technique, which allows the realization of atomically precise quantum structures, is presented. We explain the origin of the formation of quantum wires, which were grown with this molecular beam epitaxy method. The emission of quantum wire lasers based on these intrinsic T-shaped structures shows a strong dependence on magnetic fields. Only weak band gap renormalization is observed for modulation doped wires when the electron density is varied. The introduction of strained layers results in an enhancement of the confinement energy, which is necessary to fully exploit the one-dimensional character of the examined quantum wires. Single as well as coupled quantum dots, and quantum dot superlattices have been successfully grown and are identified using micro-photoluminescence spectroscopy. By combining self-assembling of quantum dots due to Stranski-Krastanov growth with the cleaved edge overgrowth method we succeeded in fabricating arrays of long-range ordered InAs quantum dots. Both atomic force microscopy and micro-photoluminescence measurements performed on such samples show the ability to control size, position, and ordering of the quantum dots.