Aims. We provide an extensive new sub-millimetre survey of the trace gas composition of Saturn's atmosphere using the broad spectral range (15-51 cm −1 ) and high spectral resolution (0.048 cm −1 ) offered by Fourier transform spectroscopy by the Herschel/SPIRE instrument (Spectral and Photometric Imaging REceiver). Observations were acquired in June 2010, shortly after equinox, with negligible contribution from Saturn's ring emission. Methods. Tropospheric temperatures and the vertical distributions of phosphine and ammonia are derived using an optimal estimation retrieval algorithm to reproduce the sub-millimetre data. The abundance of methane, water and upper limits on a range of different species are estimated using a line-by-line forward model. Results. Saturn's disc-averaged temperature profile is found to be quasi-isothermal between 60 and 300 mbar, with uncertainties of 7 K due to the absolute calibration of SPIRE. Modelling of PH 3 rotational lines confirms the vertical profile derived in previous studies and shows that negligible PH 3 is present above the 10-to 20-mbar level. The upper tropospheric abundance of NH 3 appears to follow a vapour pressure distribution throughout the region of sensitivity in the SPIRE data, but the degree of saturation is highly uncertain. The tropospheric CH 4 abundance and Saturn's bulk C/H ratio are consistent with Cassini studies. We improve the upper limits on several species (H 2 S, HCN, HCP and HI); provide the first observational constraints on others (SO 2 , CS, methanol, formaldehyde, CH 3 Cl); and confirm previous upper limits on HF, HCl and HBr. Stratospheric emission from H 2 O is suggested at 36.6 and 38.8 cm −1 with a 1σ significance level, and these lines are used to derive mole fractions and column abundances consistent with ISO and SWAS estimations a decade earlier.