We use 3035 Herschel-SPIRE 500 μm sources from 20.3 deg2 of sky in the HerMES Lockman, ES1 and XMM-LSS areas to estimate the star formation rate density at z = 0–6. 500 μm sources are associated first with 350 and 250 μm sources, and then with Spitzer 24 μm sources from the SWIRE photometric redshift catalogue. The infrared and submillimetre data are fitted with a set of radiative-transfer templates corresponding to cirrus (quiescent) and starburst galaxies. Lensing candidates are removed via a set of colour–colour and colour–redshift constraints. Star formation rates are found to extend from <1 to 20 000 M⊙ yr−1. Such high values were also seen in the all-sky IRAS Faint Source Survey. Star formation rate functions are derived in a series of redshift bins from 0 to 6, combined with earlier far-infrared estimates, where available, and fitted with a Saunders et al (1990) functional form. The star formation rate density as a function of redshift is derived and compared with other estimates. There is reasonable agreement with both infrared and ultraviolet estimates for z < 3, but we find higher star formation rate densities than ultraviolet estimates at z = 3–6. Given the considerable uncertainties in the submillimetre estimates, we cannot rule out the possibility that the ultraviolet estimates are correct. But the possibility that the ultraviolet estimates have seriously underestimated the contribution of dust-shrouded star formation can also not be excluded.