Recently, ring-shaped self-assembled InAs/GaAs nanostructures have become available [1]. Our X-STM data indicate that self-assembled InGaAs/GaAs nano-volcanoes are characterized by an asymmetric rim and a depression rather than an opening at the center [2]. We show that these asymmetric singly connected structures still can effectively manifest the electronic properties, like the Aharonov-Bohm oscillations, peculiar to doubly connected geometry. By modelling the self-assembled nano-volcanoes using the structural information from our X-STM data, we demonstrate that the occurrence of an oscillatory persistent current is surprisingly robust against such deviations from a perfect ring. We included strain and piezo-electric effects and predicted the Aharonov-Bohm oscillations of the persistent current in these non-ideal quantum rings. We report on the observation of oscillatory persistent currents in self-assembled InAs/GaAs quantum rings with a diameter less than 25 nm. In order to observe an oscillatory persistent current we measured the magnetic moment on a sample consisting of 29 layers of quantum rings, designed such that each quantum ring confines one or two electrons. We performed the magnetic moment measurements at 1.2 K and 4.2 K using an ultra-sensitive torsion magnetometer in magnetic fields up to 15 T and we were able to observe an oscillatory persistent current as large as about 60% of the persistent current in an ideal quantum ring. Both the shape and magnitude of the magnetic moment signal and the magnetic field at which the jump in the magnetic moment occurs as observed in our experiments are in good agreement with theoretical predictions. This work was been supported by the GOA BOF UA 2000, IUAP, FWO-V projects G.0274.01N, G.0435.03, the WOG WO.035.04N (Belgium), MCYT NANOSELF project (Spain) and the EC SANDiE Network of Excellence.