Most bulge-dominated galaxies host black holes (BHs) with masses that tightly correlate with the masses of their bulges. This may indicate that the BHs may regulate galaxy growth, or vice versa, or that they may grow in lock-step. The quest to understand how, when, and where those BHs formed motivates much of extragalactic astronomy. Here we focus on a population of galaxies with active BHs in their nuclei (active galactic nuclei [AGN]), that are fully or partially hidden by dust and gas: The emission from the broad line region is either completely or partially obscured with a visual extinction of one or above. This limit, though not yet precise, appears to be the point at which the populations of AGN may evolve differently. We highlight the importance of finding and studying those dusty AGN at redshifts between one and three, the epoch when the universe may have gone through its most dramatic changes. We emphasize the need for future large multiplexed spectroscopic instruments that can perform dedicated surveys in the optical and near-infrared to pin down the demographics of such objects, and study their reddening properties, star-formation histories, and excitation conditions. These key studies will shed light on the role of BHs in galaxy evolution during the epoch of peak growth activity.