Effects of Al and N codoping on the optical properties of Zn1-xMgxO in ultraviolet and visual spectral regions have been systematically investigated using density functional theory based on the first-principles method. It is found that the optical properties of Zn1-xMgxO are caused by the codoped Al and N to vary mainly in a low energy region, whereas they remain almost unchanged in a high energy region. The detailed calculations, including the imaginary part of the dielectric constant, absorption spectrum and extinction coefficient, indicate that due to the codoped Al and N, the optical absorption of Zn1-xMgxO shifts toward lower energy and the absorption of the ultraviolet and the visible absorption increase. Study on the real part of the dielectric constant and reflection spectrum of Zn1-xMgxO shows that the codoped Al and N are responsible for the increased reflection peak intensity, and also for the static dielectric constant increasing from 2.64 to 3.23. In addition, the calculations from the current electron energy loss function indicate that the codoped Al and N lead to the enhanced amplitude and the blue shift of the plasma resonant frequency of Zn1-xMgxO.