  Critical Point Symmetries (CPS) appear in regions of the nuclear chart where
a rapid change from one symmetry to another is observed. The first CPSs,
introduced by F. Iachello, were E(5), which corresponds to the transition from
vibrational [U(5)] to gamma-unstable [O(6)] behaviour, and X(5), which
represents the change from vibrational [U(5)] to prolate axially deformed
[SU(3)] shapes. These CPSs have been obtained as special solutions of the Bohr
collective Hamiltonian. More recent special solutions of the same Hamiltonian,
to be described here, include Z(5) and Z(4), which correspond to maximally
triaxial shapes (the latter with ``frozen'' gamma=30 degrees), as well as X(3),
which corresponds to prolate shapes with ``frozen'' gamma=0. CPSs have the
advantage of providing predictions which are parameter free (up to overall
scale factors) and compare well to experiment. However, their mathematical
structure [with the exception of E(5)] needs to be clarified.
