To meet the 2 °C climate target, deep cuts in greenhouse gas (GHG) emissions will be required for carbon dioxide from fossil fuels but, most likely, also for methane and nitrous oxide from agriculture and other sources. However, relatively little is known about the GHG mitigation potential in agriculture, in particular with respect to the combined effects of technological advancements and dietary changes. Here, we estimate the extent to which changes in technology and demand can reduce Swedish food-related GHG emissions necessary for meeting EU climate targets. This analysis is based on a detailed representation of the food and agriculture system, using 30 different food items. We find that food-related methane and nitrous oxide emissions can be reduced enough to meet the EU 2050 climate targets. Technologically, agriculture can improve in productivity and through implementation of specific mitigation measures. Under optimistic assumptions, these developments could cut current food-related methane and nitrous oxide emissions by nearly 50%. However, also dietary changes will almost certainly be necessary. Large reductions, by 50% or more, in ruminant meat (beef and mutton) consumption are, most likely, unavoidable if the EU targets are to be met. In contrast, continued high per-capita consumption of pork and poultry meat or dairy products might be accommodated within the climate targets. High dairy consumption, however, is only compatible with the targets if there are substantial advances in technology. Reducing food waste plays a minor role for meeting the climate targets, lowering emissions only by an additional 1–3%. © 2016 The Authors