There is an increased demand for tuna meat to supply the growing global sushi and sashimi market. Current methods to farm tunas, especially Atlantic bluefin tunas (ABFTs; Thunnus thynnus), are unsustainable. This is driving a movement towards closed-cycle ABFT aquaculture, i.e., farming tuna from spawning and hatching to slaughter on land, to try and meet the demand for these fish. Tunas are not domesticated species and thus face different challenges in terms of closed-cycle aquaculture when compared to other farmed fish species. Tunas also possess unique physiological traits such as regional endothermy and ram ventilation which affect their needs and how they are farmed. No current global standards exist to measure or monitor ABFT welfare in captivity, because there are many unknown factors surrounding ABFT welfare in aquaculture settings. There is a need to conduct studies that measure the baseline physiological parameters of ABFT in closed-cycle systems. Issues such as manipulation of breeding cycles, tank design, and slaughter procedures need attention and further research is required before such welfare indicators can be established. Stress is an overarching concern for animal welfare. In particular, pre-slaughter crowding and stress negatively affect the quality of the meat produced due to lactic acid buildup in the red muscle. Further research needs to be conducted throughout all life stages in terms of both animal welfare implications and the commercial viability of farmed ABFTs.