Assuming a continuous dependence of the criticality on the energy parameters, it is conjectured that the Ashkin-Teller model has in general two phase transitions. The two critical points coalesce into a single one when the two middle energies in the model are degenerate.
Two nonperturbative methods by means of the transfer-matrix finite-size-scaling and Monte Carlo simulations are used to investigate the spin-1 Ashkin-Teller model (ATM). We have obtained rich phase diagrams with first- and second-order phase transitions with several multicritical points of higher order. Also this model exhibits a partially ordered phase PO2 which does not exist in the spin-1/2 Ashkin-Teller model (ATM). Finally, the critical behavior of this model is discussed.