In this paper we discuss transformation properties of the spin-1 Blume-Emery-Griffiths (BEG) model arising in statistical physics. It is shown that the BEG model can always be transformed into either a spin-(1/2) Ising model or a 3-state Potts model. It is also shown that there exist other transformations relating the BEG model with spin-(1/2) king models, valid in special subspaces of the parameter space. Physical properties of the BEG model that can be drawn from the equivalence are discussed. We also present a formulation of the BEG model relating it to a 3-state vertex model.