An adequate modeling of the self-organized quantum rings is possible only on the basis of the modern characterization of those nanostructures. We discuss an atomic-scale analysis of the indium distribution of self-organized InGaAs quantum rings (QRs). The analysis of the shape, size and composition of self-organized InGaAs QRs at the atomic scale reveals that AFM only shows the material coming out of the QDs during the QR formation. The remaining QD material, as observed by Cross-Sectional Scanning Tunneling Microscopy (X-STM), shows an asymmetric indium-rich crater-like shape with a depression rather than an opening at the center and determines the observed ring-like electronic properties of QR structures. A theoretical model of the geometry and materials properties of the self-organized QRs is developed on that basis and the magnetization is calculated as a function of the applied magnetic field. Although the real QR shape differs strongly from an idealized circular-symmetric open-ring structure, Aharonov-Bohm-type oscillations in the magnetization have been predicted to survive. They have been observed using the torsion magnetometry on InGaAs QRs. Large magnetic moments of QRs are shown to originate from dissipationless circulating currents in the ground state of an electron or hole in the QR. Fine structure splitting of excitons in similar structures is shown to depend on the shape, with surprising cancellations emerging from the interplay of different contributions to the exchange; zero fine structure splitting is found for highly asymmetric structures. Examples of prospective applications of QRs are presented that do and do not utilize the topological properties of QRs.