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We calculate the excitonic transition energies and exciton binding energies in ZnO/MgxZn1-xO quantum-well heterostructures with Mg composition x varied from 0.08 to 0.36. The effect of the exciton-phonon interaction on the exciton binding energies is taken into account in the model. For the ZnO/Mg0.12Zn0.88O quantum-well structure, we compare the calculated result with the available experimental data at 5 K, and a good agreement is achieved. The excitonic transition energies at room temperature in ZnO/MgxZn1-xO quantum-well heterostructures are also calculated. The results show that when the well width exceeds 50 A, the quantum size effect is neglectable and the excitonic transition energies in ZnO/MgxZn1-xO (with x varied from 0.08 to 0.36)quantum-well heterostructures are close to the value of bulk ZnO. The maximum exciton binding energy as large as 121.1 meV is obtained for the well width of 12.5 A in the ZnO/Mg0.36Zn0.64O quantum-well heterostructures.