Tuna fisheries have been identified as one of the major threats to populations of other marine vertebrates, including sea turtles, sharks, seabirds and marine mammals. The development of technical mitigation measures (MM) in fisheries is part of the code of conduct for responsible fisheries. An in-depth analysis of the available literature regarding bycatch mitigation in tuna fisheries with special reference to elasmobranchs was undertaken. Studies highlighting promising MMs were reviewed for four tuna fisheries (longline, purse seine, driftnets and gillnet, and rod and line – including recreational fisheries). The advantages and disadvantages of different MMs are discussed and assessed based on current scientific knowledge. Current management measures for sharks and rays in tuna Regional Fishery Management Organizations (t-RFMOs) are presented. A review of relevant studies examining at-vessel and postrelease mortality of elasmobranch bycatch is provided. This review aims to help fisheries managers identify pragmatic solutions to reduce mortality on pelagic elasmobranchs (and other higher vertebrates) whilst minimizing impacts on catches of target tuna species. Recent research efforts have identified several effective MMs that, if endorsed by t-RFMOs, could reduce elasmobranchs mortality rate in international tropical purse seine tuna fisheries. In the case of longline fisheries, the number of operational effective MMs is very limited. Fisheries deploying driftnets in pelagic ecosystems are suspected to have a high elasmobranchs bycatch and their discard survival is uncertain, but no effective MMs have been field validated for these fisheries. The precautionary bans of such gear by the EU and by some t-RFMOs seem therefore appropriate. Recreational tuna fisheries should be accompanied by science-based support to reduce potential negative impacts on shark populations. Priorities for research and management are identified and discussed.