  Prolate to oblate shape transitions have been predicted in an analytic way in
the framework of the Interacting Boson Model (IBM), determining O(6) as the
symmetry at the critical point. Parameter-independent predictions for prolate
to oblate transitions in various regions on the nuclear chart have been made in
the framework of the proxy-SU(3) and pseudo-SU(3) symmetries, corroborated by
recent non-relativistic and relativistic mean field calculations along series
of nuclear isotopes, with parameters fixed throughout, as well as by shell
model calculations taking advantage of the quasi-SU(3) symmetry. Experimental
evidence for regions of prolate to oblate shape transitions is in agreement
with regions in which nuclei bearing the O(6) dynamical symmetry of the IBM
have been identified, lying below major shell closures. In addition, gradual
oblate to prolate transitions are seen when crossing major nuclear shell
closures, in analogy to experimental observations in alkali clusters.
