To assimilate atmospheric and surface radiance measurements from satellites in a numerical weather prediction model, a fast radiative transfer model is required to compute radiances from the model first guess fields at every observation point. Such a model for satellite infrared and microwave radiance measurements is used operationally for the assimilation of TIROS operational vertical sounder radiances at the European Centre for Medium-Range Weather Forecasts. An improved version of this model has been developed which requires ozone, in addition to temperature and water vapour, in the input profile and it has been generalized to compute radiances for other satellite radiometers using the same code. Instruments such as the high resolution infrared radiation sounder and the advanced microwave sounding unit on the National Oceanic and Atmospheric Administration polar orbiters, the METEOSAT water vapour imager and the Geostationary Operational Environmental Satellite infrared sounder have been simulated. It is demonstrated, by comparisons with line-by-line model computed radiances, that the fast model can reproduce the line-by-line model radiances for the TIROS operational vertical sounder stratospheric temperature sounding channels to an accuracy below the instrumental noise. The tropospheric, surface sensing, water vapour and ozone channel radiances cannot be predicted to such an accuracy, but still accurately enough for numerical weather prediction assimilation. A comparison of measured TIROS operational vertical sounder radiances with predicted values from numerical weather prediction model analyses gives larger differences than would be expected from the combination of the fast model and instrument related errors for most channels. The validity of the tangent linear approximation of the model gradient for typical radiance departures is also explored, with several examples, for the high resolution infrared radiation sounder/advanced microwave sounding unit instrument combination. The tangent-linear approximation is valid for temperature but significant departures from linearity about the first guess profile are observed for water vapour and ozone. Cloud affected infrared radiances have a highly non-linear response.