Recent progress in the fabrication of GaAs/AlGaAs low-dimensional structures by cleaved edge overgrowth (CEO) — a molecular beam growth technique that involves one or two regrowth steps on the sidewalls of an in situ cleaved layer structure — is reviewed. Ballistic electron transport in modulation-doped quantum wires prepared in this way is characterized by pronounced quantization of the conductance in integer multiples < 2e2/h. A magnetic field oriented perpendicular to the quantum wires is found to increase the quantized conductance values. While for a 400 Å wide channel the canonical values of n × 2e2/h are almost reached at a magnetic field strength of 4 T the corresponding conductance rise in a 250 Å wide channel is much slower. In addition, the magnetic field dependence of the positions of the conductance steps as a function of an applied gate voltage which controls the electron density is distinctly different for wires of 400 and 250 Å width. The optical properties of atomically precise quantum dots which originate at the right angle intersection of three quantum wells are characterized by narrow “atom-like” emission lines. Two different types of quantum dot structures have been investigated using microscopic photoluminescence spectroscopy: linear arrays of weakly coupled dots, i.e. an one-dimensional quantum dot superlattice and single as well as paired quantum dots. The latter allow the assembly of “artificial molecules” out of almost identical quantum dots which can be considered as “artificial atoms” as the dot-to-dot distance is varied.