  Theoretical approaches to one-dimensional and quasi-one-dimensional quantum
rings with a few electrons are reviewed. Discrete Hubbard-type models and
continuum models are shown to give similar results governed by the special
features of the one-dimensionality. The energy spectrum of the many-body states
can be described by a rotation-vibration spectrum of a 'Wigner molecule' of
'localized' electrons, combined with the spin-state determined from an
effective antiferromagnetic Heisenberg Hamiltonian. The persistent current as a
function of the magnetic flux through the ring shows periodic oscillations
arising from the 'rigid rotation' of the electron ring. For polarized electrons
the periodicity of the oscillations is always the flux quantum Phi_0. For
nonpolarized electrons the periodicity depends on the strength of the effective
Heisenberg coupling and changes from Phi_0 first to Phi_0/2 and eventually to
Phi_0/N when the ring gets narrower.
