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We present Keck spectroscopic observations and redshifts for a sample of 767Herschel-SPIRE selected galaxies (HSGs) at 250, 350, and 500um, taken with theKeck I Low Resolution Imaging Spectrometer (LRIS) and the Keck II DEep ImagingMulti-Object Spectrograph (DEIMOS). The redshift distribution of these SPIREsources from the Herschel Multitiered Extragalactic Survey (HerMES) peaks atz=0.85, with 731 sources at z<2 and a tail of sources out to z~5. We measuremore significant disagreement between photometric and spectroscopic redshifts(<delta_z>/(1+z)>=0.29) than is seen in non-infrared selected samples, likelydue to enhanced star formation rates and dust obscuration in infrared-selectedgalaxies. We estimate that the vast majority (72-83%) of z<2 Herschel-selectedgalaxies would drop out of traditional submillimeter surveys at 0.85-1mm. Weestimate the luminosity function and implied star-formation rate densitycontribution of HSGs at z<1.6 and find overall agreement with work based on24um extrapolations of the LIRG, ULIRG and total infrared contributions. Thiswork significantly increased the number of spectroscopically confirmedinfrared-luminous galaxies at z>>0 and demonstrates the growing importance ofdusty starbursts for galaxy evolution studies and the build-up of stellar massthroughout cosmic time. [abridged]

A Redshift Survey of Herschel Far-Infrared Selected Starbursts and Implications for Obscured Star Formation