  We investigate of the properties of $\sim$2000 Herschel/SPIRE-selected
galaxies from $0<z<4$ using a combination of extensive spectroscopy, deep
imaging from CFHT, VLA, Spitzer, XMM-Newton, and Herschel, and well-calibrated
SED fitting. Herschel galaxies are observed to span a range of stellar masses,
colors, and absolute magnitudes equivalent to galaxies undetected in SPIRE.
Though many Herschel galaxies appear to be in transition, such galaxies are
largely consistent with normal star-forming galaxies when rest-frame colors are
utilized. The nature of the star-forming "main sequence" is studied and we warn
against adopting this framework unless the main sequence is determined
precisely. Herschel galaxies at different total infrared luminosities
($L_{TIR}$) are compared. Bluer colors, larger nebular extinctions, and larger
contributions from younger stellar populations are observed for galaxies with
larger $L_{TIR}$, suggesting that low-$L_{TIR}$ galaxies are undergoing
rejuvenated starbursts while galaxies with higher $L_{TIR}$ are forming a
larger percentage of their stellar mass. A variety of methods are used to
select powerful active galactic nuclei (AGN). Galaxies hosting AGN are observed
to be undergoing starbursts more commonly and vigorously than a matched sample
of galaxies without powerful AGN and, additionally, the fraction of galaxies
with an AGN increases with increasing star formation rate at all redshifts. At
all redshifts ($0<z<4$) the most prodigious star-forming galaxies are found to
contain the highest fraction of powerful AGN. For redshift bins that allow a
comparison ($z>0.5$), the highest $L_{TIR}$ galaxies in a given redshift bin
are unobserved by SPIRE at subsequently lower redshifts, a trend linked to
downsizing. In conjunction with other results, this evidence is used to argue
for prevalent AGN-driven quenching in starburst galaxies across cosmic time.
