  We have conducted a long slit search for low surface brightness Lyman-alpha
emitters at redshift 2.67 < z < 3.75. A 92 hour long exposure with VLT/FORS2
down to a 1-sigma surface brightness detection limit of 8x10^-20
erg/cm2/s/sqarcsec yielded a sample of 27 single line emitters with fluxes of a
few times 10^-18 erg/s/cm2. We present arguments that most objects are indeed
Lyman-alpha. The large comoving number density, the large covering factor,
dN/dz ~ 0.2-1, and the often extended Lyman-alpha emission suggest that the
emitters be identified with the elusive host population of damped Lyman-alpha
systems (DLAS) and high column density Lyman limit systems. A small inferred
star formation rate, perhaps supplanted by cooling radiation, appears to
energetically dominate the Lyman-alpha emission, and is consistent with the low
metallicity, low dust content, and theoretically inferred low masses of DLAS,
and with the relative lack of success of earlier searches for their optical
counterparts. (abridged)
