In this paper I review current knowledge of the atmospheres of Jupiter and Saturn, making use of the extensive telescopic studies, International Ultraviolet Explorer Satellite observations and the measurements made during the recent Pioneer and Voyager flybys which have been supported by detailed theoretical studies. A detailed discussion is given of the composition of these atmospheres and the abundance ratios which provide insight into their original state and their evolution. The Voyager observations indicate a surprisingly close similarity between the weather systems of the Earth and the giant planets. Although both Jupiter and Saturn have internal heat sources, and are therefore star-like in their interiors, they appear to produce terrestrial-style weather systems. A detailed discussion is given of this work, which forms a major study of the Laboratory for Planetary Atmospheres at University College London.