Efficient and flexible algorithms for the retrieval of atmospheric constituent and aerosol profiles from a limb-sounding satellite radiometer are presented. The radiative transfer scheme ( “forward model”) has been generalized to cope with departures from local thermodynamic equilibrium (LTE), modelling emission from fundamental bands as well as hot bands. Its formulation makes possible a fast method of accurately calculating the radiance derivatives (“weighting functions”) required for the nonlinear optimal estimation algorithms. The retrieval schemes can be used with a wide variety of signal to noise ratios and with measurements from more than one wavelength. Multiple retrieval products are possible. A complete error analysis method emerges naturally from the theory, permitting an extensive prelaunch characterization of the retrieval products. The algorithms have been implemented for the improved stratospheric and mesospheric sounder (ISAMS), an infrared limb sounder on the Upper Atmosphere Research Satellite. The retrieval algorithms and error analysis are illustrated with some of the prelaunch tests performed for ISAMS. The O3 retrieval exemplifies those using measurements with a very high signal to noise ratio for which the quality of the retrieved product will be limited by spectroscopic and other forward model deficiencies. Preliminary calculations indicate that it should be possible to perform retrievals of daytime CO in the mesosphere and lower thermosphere because of rather than in spite of the strong non-LTE emission in that region due to solar pumping at 4.6 μm. The “optimal onion-peeling” algorithm used in the joint retrieval of CH4 and N2O is a very accurate means of solving strong cross contamination problems and proves to be much quicker than any iterative sequence of single-product retrievals.