The deep sea is the largest ecosystem on Earth but organisms living there must contend with high pressure, low temperature, darkness and scarce food. Chondrichthyan fishes (sharks and their relatives) are important consumers in most marine ecosystems but are uncommon deeper than 3000 m and exceedingly rare, or quite possibly absent, from the vast abyss (depths >4000 m). By contrast, teleost (bony) fishes are commonly found to depths of ∼8400 m. Why chondrichthyans are scarce at abyssal depths is a major biogeographical puzzle. Here, after outlining the depth-related physiological trends among chondrichthyans, we discuss several existing and new hypotheses that implicate unique physiological and biochemical characteristics of chondrichthyans as potential constraints on their depth distribution. We highlight three major, and not mutually exclusive, working hypotheses: (1) the urea-based osmoregulatory strategy of chondrichthyans might conflict with the interactive effects of low temperature and high pressure on protein and membrane function at great depth; (2) the reliance on lipid accumulation for buoyancy in chondrichthyans has a unique energetic cost, which might increasingly limit growth and reproductive output as food availability decreases with depth; (3) their osmoregulatory strategy may make chondrichthyans unusually nitrogen limited, a potential liability in the food-poor abyss. These hypotheses acting in concert could help to explain the scarcity of chondrichthyans at great depths: the mechanisms of the first hypothesis may place an absolute, pressure-related depth limit on physiological function, while the mechanisms of the second and third hypotheses may limit depth distribution by constraining performance in the oligotrophic abyss, in ways that preclude the establishment of viable populations or lead to competitive exclusion by teleosts.