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 PH3 rotational lines confirms the vertical profile derived in previous studies and shows that negligible PH3 is present above the 10- to 20-mbar level. The upper tropospheric abundance of NH3 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 CH4 abundance and Saturn's bulk C/H ratio are consistent with Cassini studies. We improve the upper limits on several species (H2S, HCN, HCP and HI); provide the first observational constraints on others (SO2, CS, methanol, formaldehyde, CH3Cl); and confirm previous upper limits on HF, HCl and HBr. Stratospheric emission from H2O 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.Read less <