Low surface brightness (LSB) galaxies 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 results from our ongoing search for LSB galaxies with the Hyper Suprime-Cam Survey, a new generation imaging survey using the 8.2-meter Subaru Telescope on Mauna Kea. We are uncovering a diverse population of objects, spanning dwarf ellipticals in nearby galaxy groups to ultra-diffuse galaxies in the field to giant LSB spirals. We are probing the distance distribution of our sample using a variety of methods including optical and HI spectroscopy, globular cluster counts, galaxy clustering, and surface brightness fluctuations. Importantly, this follow-up work will enable us to study the numbers and physical properties of these objects 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.