SPIRE, the Spectral and Photometric Imaging Receiver, is one of three instruments to fly on the European Space Agency's Herschel Space Observatory. It contains a three-band imaging photometer operating at 250, 350 and 500 μm, and an imaging Fourier transform spectrometer covering 194-672 μm. The SPIRE detectors are arrays of feedhorn-coupled bolometers cooled to 0.3 K. The photometer has a field of view of 4 × 8′, observed simultaneously in the three spectral bands. The spectrometer has an approximately circular field of view with a diameter of 2.6′ The spectral resolution can be adjusted between 0.04 and 2 cm-1 (resolving power of 20-1000 at 250 μm). SPIRE will be used for many galactic and extragalactic science programmes, a number of which will be implemented as Herschel Key Projects. The SPIRE consortium's Guaranteed Time (GT) programme will devote more than 1000 h to Key Projects covering the high-redshift universe and local galaxies, which will be carried out in coordination with other GT programmes, especially that of the PACS consortium. It is also expected that substantial amounts of Herschel Open Time will be used for further extragalactic investigations. The high-redshift part of the SPIRE GT programme will focus on blank-field surveys with a range of depths and areas optimised to sample the luminosity-redshift plane and characterize the bolometric luminosity density of the universe at high-redshift. Fields will be selected that are well covered by Spitzer, SCUBA-2, PACS-GT and near-IR surveys, to facilitate source identifications and enable detailed studies of the redshifts, spectral energy distributions, and other properties of detected galaxies. The local galaxies programme will include a detailed spectral and photometric study of a sample of well resolved nearby galaxies, a survey of more than 300 local galaxies designed to provide a statistical survey of dust in the nearby universe, and a study of the ISM in low-metallicity environments, bridging the gap between the local universe and primordial galaxies.