Study of the nuclear shape phase transitions and of the corresponding critical points using different approaches as Bohr Model, Interacting Boson Model and Coherent State Model. The attention is focused more on the low-lying quadrupole collective states defining the ground band and the first beta and gamma bands. A special interest is accorded to the critical point of the phase transition from spherical shape to deformed one, where the shape is not clearly defined. In this point, depending on deformation, one can find shape fluctuations, mixing and coexistence, respectively.

It is well established the existence of the pairing interaction in the nuclear physics. A new more general pairing mechanism is further investigated, known as quartet or alphalike four-body structure. The later implies the coupling of two protons and two neutrons to total isospin T=0. The model, which involves such couplings, is known in literature as Quartet Condensation Model.