Thesdg interacting boson model (sdgIBM), which includes monopole (s), quadrupole (d) and hexadecupole (g) degrees of freedom, enables one to analyze hexadecupole (E4) properties of atomic nuclei. Various aspects of the model, both analytical and numerical, are reviewed emphasizing the symmetry structures involved. A large number of examples are given to provide understanding and tests, and to demonstrate the predictiveness of thesdg model. Extensions of the model to include proton-neutron degrees of freedom and fermion degrees of freedom (appropriate for odd mass nuclei) are briefly described. A comprehensive account ofsdgIBM analysis of all the existing data on hexadecupole observables (mainly in the rare-earth region) is presented, includingβ4 (hexadecupole deformation) systematics,B(IS4; 0GS+→4γ+) systematics that give information about hexadecupole component in γ-vibration,E4 matrix elements involving few low-lying 4+ levels,E4 strength distributions and hexadecupole vibrational bands in deformed nuclei.