We present our current understanding of the composition, vertical mixing, cloud structure and the origin of the atmospheres of Jupiter and Saturn. Available observations point to a much more vigorous vertical mixing in Saturn's middle-upper atmosphere than in Jupiter's. The nearly cloud-free nature of the Galileo probe entry site, a 5-micron hotspot, is consistent with the depletion of condensable volatiles to great depths, which is attributed to local meteorology. Somewhat similar depletion of water may be present in the 5-micron bright regions of Saturn also. The supersolar abundances of heavy elements, particularly C and S in Jupiter's atmosphere and C in Saturn's, as well as the progressive increase of C from Jupiter to Saturn and beyond, tend to support the icy planetesimal model of the formation of the giant planets and their atmospheres. However, much work remains to be done, especially in the area of laboratory studies, including identification of possible new microwave absorbers, and modelling, in order to resolve the controversy surrounding the large discrepancy between Jupiter's global ammonia abundance, hence the nitrogen elemental ratio, derived from the earth-based microwave observations and that inferred from the analysis of the Galileo probe-orbiter radio attenuation data for the hotspot. We look forward to the observations from Cassini-Huygens spacecraft which are expected to result not only in a rich harvest of information for Saturn, but a better understanding of the formation of the giant planets and their atmospheres when these data are combined with those that exist for Jupiter.