To widen the band gap of ZnO, we have investigated $Zn_{1-x}Mg_xO(ZMO)$ thin films prepared by pulsed laser deposition on c-plane sapphire substrates at $500^{circ}C$. From X-ray diffraction patterns, ZMO films show only the (0002) and (0004) diffraction peaks. It means that the flints have the wurtzite structure. Segregation of ZnO and MgO phases is found in the films with x=0.59. All the samples are highly transparent in the visible region and have a sharp absorption edge in the UV region. The shift of absorption edge to higher energy is observed in the films with higher Mg composition. The excitonic nature of the films is clearly appeared in the spectra for all alloy compositions. The optical band-gap ($E_g$) of ZMO films is obtained from the ${alpha}^2$ vs Photon energy plot assuming ${alpha}^2;propto$ (hv - $E_g$), where u is the absorption coefficient and hv is the photon energy. The value of $E_g$ increases up to 3.72 eV for the films with x=0.35. It is important to adjust Mg composition control for controlling the band-gap of ZMO films.