  Studies of the Interacting Boson Approximation (IBA) model for large boson
numbers have been triggered by the discovery of shape/phase transitions between
different limiting symmetries of the model. These transitions become sharper in
the large boson number limit, revealing previously unnoticed regularities,
which also survive to a large extent for finite boson numbers, corresponding to
valence nucleon pairs in collective nuclei. It is shown that energies of 0_n^+
states grow linearly with their ordinal number n in all three limiting
symmetries of IBA [U(5), SU(3), and O(6)]. Furthermore, it is proved that the
narrow transition region separating the symmetry triangle of the IBA into a
spherical and a deformed region is described quite well by the degeneracies
E(0_2^+)=E(6_1^+), E(0_3^+)=E(10_1^+), E(0_4^+)=E(14_1^+), while the energy
ratio E(6_1^+) /E(0_2^+) turns out to be a simple, empirical, easy-to-measure
effective order parameter, distinguishing between first- and second-order
transitions. The energies of 0_n^+ states near the point of the first order
shape/phase transition between U(5) and SU(3) are shown to grow as n(n+3), in
agreement with the rule dictated by the relevant critical point symmetries
resulting in the framework of special solutions of the Bohr Hamiltonian. The
underlying partial dynamical symmetries and quasi-dynamical symmetries are also
discussed.
