An overview is given on the recent experimental and theoretical advancements in studies of novel manifestations of the Aharonov-Bohm quantum-interference effect for excitons confined to selfassembled quantum rings and other semiconductor nanostructures with ring-like states of charge carriers as well as for electrons in Möbius rings at the micro- and nanoscale. The exciton Aharonov-Bohm effect can be effectively controlled by an out-of-plane magnetic field, a vertical electric field, a spin disorder. A “delocalization-to-localization” transition for the electron ground state occurs in a Möbius ring as it is made more inhomogeneous.