  We introduce a very deep, R_lim~27, multicolor imaging survey of very faint
star-forming galaxies at z~4, z~3, z~2.2, and z~1.7. This survey, carried out
on the Keck I telescope, uses the very same UGRI filter system that is employed
by the Steidel team to select galaxies at these redshifts, and thus allows us
to construct identically-selected, but much fainter, samples. However, our
survey reaches ~1.5 mag deeper than the work of Steidel and his group, letting
us probe substantially below the characteristic luminosity L* and thus study
the properties and redshift evolution of the faint component of the high-z
galaxy population. The survey covers 169 square arcminutes in three spatially
independent patches on the sky and -- to R<~27 -- contains 427 GRI-selected z~4
LBGs, 1481 UGR-selected z~3 LBGs, 2417 UGR-selected z~2.2 star-forming
galaxies, and 2043 UGR-selected z~1.7 star-forming galaxies. In this paper, the
first in a series, we introduce the survey, describe our observing and data
reduction strategies, and outline the selection of our z~4, z~3, z~2.2, and
z~1.7 samples.
