Recent progress in the fabrication of atomically precise quantum wires and quantum dots by cleaved edge overgrowth is reviewed. This technique is based on molecular beam epitaxy along two or three directions in space and is capable of producing low-dimensional structures in which all relevant sizes are controlled on an atomic scale. Different sample structures consisting of isolated quantum dots as well as of linear arrays of coupled quantum dots are studied by micro photoluminescence spectroscopy. Optical emission from zero-dimensional states in these dots is characterized by extremely narrow linewidths of less than 70 eV. Resonant excitation of one of the dot levels in coupled systems leads to emission from all the other levels. As a result of the quantum mechanical coupling of the almost identical dots anti-Stokes luminescence, i.e. thermally activated transfer of oscillator strength from one dot to another, whose emission is slightly higher in energy, is observed.