Low-surface-brightness galaxies (LSBGs) are a significant component of the galaxy population, which provide a unique testing ground for theoretical predictions of galaxy and star formation, stellar feedback processes, and the distribution and nature of dark matter. However, their defining characteristic—central surface brightnesses that are fainter than the night sky—makes them difficult to detect and study, leading to their underrepresentation in previous optical surveys and biasing our view of the full galaxy population. I will present a new view of these elusive galaxies from the Hyper Suprime-Cam (HSC) Survey, an ambitious 300-night imaging survey using the 8.2-meter Subaru Telescope on Mauna Kea. After giving an overview of the HSC Survey, I will present our source-detection pipeline and initial catalog of LSBGs within the first ~200 deg^2 of the survey, which will grow to 1400 deg^2 upon survey completion. Our LSBG catalog will facilitate follow-up efforts (which we have already started) to study the physical properties and number densities of these galaxies as a function of environment. Pushing such studies to lower surface brightnesses will be necessary to form a more complete census of the galaxy population, which will ultimately provide one of the strongest tests of the standard LCDM framework.