Damped Ly-alpha Absorbers (DLAs) enable studies of the baryon cycle because they probe the physical properties of high column density HI gas beyond z=1. Although connecting DLAs with the population of galaxies detected in emission has been a challenge for decades, this is now changing with the advent of new technologies. In this talk, I will introduce the largest search for the galaxies associated with DLAs to date: a multi-year survey utilizing the integral field spectrograph KCWI at Keck. So far, we have detected ~20 LAEs within 100 projected kpc of ~45 DLAs at z=2-5, corresponding to a ~40% detection rate. We find that LAEs are often at large impact parameters (> 40 kpc) and that they are typically associated with DLAs of low-intermediate metallicities (-2.5 < [M/H] < -0.5). Our preferred interpretation for these results is that moderately enriched DLAs probe the circum-galactic medium of low mass (<10^10 Msun) galaxies. On the other hand, metal-rich DLAs are often associated with massive galaxies that are faint in Ly-alpha and bright in the sub-millimeter (e.g. CO and CII emitters). Our survey highlights that complementary approaches are necessary to identify the wide range of galaxy types associated with DLAs.