ZnO films were grown by radical-source molecular beam epitaxy (RS-MBE) on sapphire and glass substrates, and they were characterized in terms of Hall mobility and optical transmission. Undoped ZnO films exhibit a low intrinsic defect density and optical properties close to bulk ZnO. By Ga doping, a resistance ρ as low as 2×10−4 Ωcm could be achieved. Balancing high conductivity and low transmission losses due to free carrier absorption in the infrared, the optimum was obtained for ρ=3.4×10−4Ωcm, electron mobility μe=37 cm2/Vs and an average transmission T of 96% in the wavelength range 400-1100 nm. Polycrystalline growth on glass yields slightly reduced but still good film quality (μe=30 cm2/Vs, T=90%). By the incorporation of Mg, conducting Mg0.3Zn0.7O films with an increased band gap up to ∼ 4eV were realized.