Recent advances in the chemical vapor deposition method of growing graphene sheets suggest that graphene rings can grow. We may anticipate that chemical methods can be developed to construct twisted nano-ribbons to form Möbius structures in the very near future. I investigated the quantum mechanics of an electron constrained to motion on a nanoscale Möbius ring by solving the Schrdinger equation on the curved surface. The close analogy between ordinary cylindrical rings and Möbius rings is displayed by the closeness of their energy spectra. The expectation values for the angular momentum component L_z are shown to be close, but not exactly equal, to integral or half-integral multiples of hbar. The half-integer angular momentum states are present only for the nontrivial topology of Möbius rings. The effect of the curvature of the Möbius rings manifests itself in the level splitting. This can be understood in terms of representations of the discrete rotational groups C_nv. The nonzero variance of L_z will allow weak transitions between integral and half-integral angular momentum states, while preserving the unit angular momentum for photons. Again, since the topology of the system is critical for the Aharonov-Bohm effect, I investigated the AB effect on Möbius rings and found a remarkable pattern in transmission through finite-width 2D ring structures with finite-width input and output contacts attached at the periphery. The periodicity in the magnetic flux, in units of h/e, is weakly broken on 2D rings of finite width. The unusual states with half-integer values of observed on Möbius rings, investigated earlier, display a different characteristic in transmission. In view of the fascinating properties displayed by the non-trivial topology in terms of its novel two-dimensional physics, we expect that the properties of carriers on the Möbius ring that we have presented here will be relevant for practical applications.