  After a brief historical introduction, we dwell on two recent experiments in
the low-temperature, monoclinic phase of V_2O_3: K-edge resonant x-ray
scattering and non-reciprocal linear dichroism, whose interpretations are in
conflict, as they require incompatible magnetic space groups. Such a conflict
is critically reviewed, in the light of the present literature, and new
experimental tests are suggested, in order to determine unambiguously the
magnetic group. We then focus on the correlated, non-local nature of the
ground-state wave function, that is at the basis of some drawbacks of the LDA+U
approach: we singled out the physical mechanism that makes LDA+U unreliable,
and indicate the way out for a possible remedy. Finally we explain, by means of
a symmetry argument related to the molecular wave function, why the magnetic
moment lies in the glide plane, even in the absence of any local symmetry at
vanadium sites.
