  We present the results of our systematic search for the reddest far-infrared
(FIR) and submillimeter (sub-mm) galaxies using the data from the Herschel
Multi-tiered Extragalactic Survey (HerMES) and the SCUBA2 Cosmological Legacy
Survey (S2CLS). The red FIR galaxies are "500~$\mu$m risers," whose spectral
energy distributions increase with wavelength across the three FIR passbands of
the Spectral and Photometric Imaging REceiver (SPIRE) of Herschel. Within 106.5
deg$^2$ of the HerMES fields, we have selected 629 500 $\mu$m risers. The red
sub-mm galaxies are "SPIRE dropouts," which are prominent detections in the
S2CLS 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 95
such objects. These very red sources could be dusty starbursts at high
redshifts ($z\gtrsim 4$-6) because the peak of their cold-dust emission heated
by star formation is shifted to the reddest FIR/sub-mm bands. The surface
density of 500 $\mu$m risers is $\sim$8.2 deg$^{-2}$ at the $\geq 20$ mJy level
in 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 range
of redshifts, however. Using deep radio data in these fields to further select
the 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 0.8-13.6
deg$^{-2}$ among SPIRE dropouts. If this is correct, the dust-embedded star
formation processes in such objects could contribute comparably as Lyman-break
galaxies to the global SFR density at $z>6$.
