We present Keck spectroscopic observations and redshifts for a sample of 767 Herschel-SPIRE selected galaxies (HSGs) at 250, 350, and 500um, taken with the Keck I Low Resolution Imaging Spectrometer (LRIS) and the Keck II DEep Imaging Multi-Object Spectrograph (DEIMOS). The redshift distribution of these SPIRE sources from the Herschel Multitiered Extragalactic Survey (HerMES) peaks at z=0.85, with 731 sources at z<2 and a tail of sources out to z~5. We measure more significant disagreement between photometric and spectroscopic redshifts (<delta_z>/(1+z)>=0.29) than is seen in non-infrared selected samples, likely due to enhanced star formation rates and dust obscuration in infrared-selected galaxies. We estimate that the vast majority (72-83%) of z<2 Herschel-selected galaxies would drop out of traditional submillimeter surveys at 0.85-1mm. We estimate the luminosity function and implied star-formation rate density contribution of HSGs at z<1.6 and find overall agreement with work based on 24um extrapolations of the LIRG, ULIRG and total infrared contributions. This work significantly increased the number of spectroscopically confirmed infrared-luminous galaxies at z>>0 and demonstrates the growing importance of dusty starbursts for galaxy evolution studies and the build-up of stellar mass throughout cosmic time. [abridged]
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