  The phenomenological classification of collective quadrupole excitations by
means of the Bohr Hamiltonian is reviewed with focus on signatures for
triaxility. The variants of the microscopic Bohr Hamiltonian derived by means
of the Adiabatic Time Dependent Mean Field theory from the Pairing plus
Quadrupole-Quadrupole interaction, the Shell Correction Method, the Skyrme
Energy Density Functional, the Relativistic Mean Field Theory, and the Gogny
interaction are discussed and applications to concrete nuclides reviewed. The
Generator Coordinate Method for the five dimensional quadrupole deformation
space and first applications to triaxial nuclei are presented. The
phenomenological classification in the framework of the Interacting Boson Model
is discussed with a critical view on the boson number counting rule. The recent
success in calculating the model parameters by mapping the mean field
deformation energy surface on the bosonic one is discussed and the applications
listed. A critical assessment of the models is given with focus on the
limitations due to the adiabatic approximation. The Tidal Wave approach and the
Triaxial Projected Shell Model are presented as practical approaches to
calculate spectral properties outside the adiabatic region.
