A comprehensive hot line list is calculated for ^31^PH_3_ in its ground electronic state. This line list, called SAlTY, contains almost 16.8 billion transitions between 7.5 million energy levels and it is suitable for simulating spectra up to temperatures of 1500K. It covers wavelengths longer than 1um and includes all transitions to upper states with energies below hc.18000cm^-1^ and rotational excitation up to J=46. The line list is computed by variational solution of the Schrodinger equation for the rotation-vibration motion employing the nuclear-motion program TROVE. A previously reported ab initio dipole moment surface is used as well as an updated 'spectroscopic' potential energy surface (PES), obtained by refining an existing ab initio surface through least-squares fitting to the experimentally derived energies. Detailed comparisons with other available sources of phosphine transitions confirms SAlTY's accuracy and illustrates the incompleteness of previous experimental and theoretical compilations for temperatures above 300K. Atmospheric models are expected to severely underestimate the abundance of phosphine in disequilibrium environments, and it is predicted that phosphine will be detectable in the upper troposphere of many substellar objects. This list is suitable for modelling atmospheres of many astrophysical environments, namely carbon stars, Y dwarfs, T dwarfs, hot Jupiters and solar system gas giant planets. It is available in full as supplementary data to the article and at www.exomol.com.