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We present the results of our systematic search for the reddest far-infrared(FIR) and submillimeter (sub-mm) galaxies using the data from the HerschelMulti-tiered Extragalactic Survey (HerMES) and the SCUBA2 Cosmological LegacySurvey (S2CLS). The red FIR galaxies are "500~$\mu$m risers," whose spectralenergy distributions increase with wavelength across the three FIR passbands ofthe Spectral and Photometric Imaging REceiver (SPIRE) of Herschel. Within 106.5deg$^2$ of the HerMES fields, we have selected 629 500 $\mu$m risers. The redsub-mm galaxies are "SPIRE dropouts," which are prominent detections in theS2CLS 850 $\mu$m data but are extremely weak or invisible in the SPIRE bands.Within the 2.98 deg$^2$ common area of HerMES and S2CLS, we have selected 95such objects. These very red sources could be dusty starbursts at highredshifts ($z\gtrsim 4$-6) because the peak of their cold-dust emission heatedby star formation is shifted to the reddest FIR/sub-mm bands. The surfacedensity of 500 $\mu$m risers is $\sim$8.2 deg$^{-2}$ at the $\geq 20$ mJy levelin 500 $\mu$m, while that of SPIRE dropouts is $\sim$19.3 deg$^{-2}$ at the$\geq 5$ mJy level in 850 $\mu$m. Both type of objects could span a wide rangeof redshifts, however. Using deep radio data in these fields to further selectthe ones likely at the highest redshifts, we find that the surface density of$z>6$ candidates is 5.5 deg$^{-2}$ among 500 $\mu$m risers and is 0.8-13.6deg$^{-2}$ among SPIRE dropouts. If this is correct, the dust-embedded starformation processes in such objects could contribute comparably as Lyman-breakgalaxies to the global SFR density at $z>6$.

A Systematic Search for Reddest Far-infrared and Sub-millimeter Galaxies: revealing dust-embedded starbursts at high redshifts