  We use 3035 Herschel-SPIRE 500$\mu$m sources from 20.3 sq deg of sky in the
HerMES Lockman, ES1 and XMM-LSS areas to estimate the star-formation rate
density at z = 1-6. 500 mu sources are associated first with 350 and 250 mu
sources, and then with Spitzer 24 mu 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 Mo/yr. 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 0-6, 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 can not 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.
