We have grown epitaxial MgxZn1-xO alloy films and ZnO/MgxZn1-xO (x = 0.20) superlattices on sapphire(0001) substrates by laser molecular beam epitaxy and characterized their structures by X-ray diffraction. Single phase MgxZn1-xO could be obtained up to x = 0.33, whereas MgO impurity phase with (111) orientation segregated at x > 0.33. The bandgap of MgxZn1-xO was successfully controlled as verified by the photoluminescence peaks shifting 3.36 eV (x = 0) to 3.87 eV (x = 0.33). It was found that the structure of the superlattices was greatly improved by the use of a ZnO buffer layer on sapphire substrate prior to the deposition of superlattice. Small angle X-ray diffraction peaks corresponding to the period of the superlattices ranging from 8 to 18 nm could be clearly observed.