Journal of Political Philosophy 2020. States are free, yet everywhere live under international courts and tribunals (ICs). As they proliferate and gain power across ever more domains, ICs become targets of resistance and criticism that they are illegitimate authorities. What reasons might a state have to defer to an IC’s judgment or interpretation, even when the state regards it as mistaken, and even when it conflicts with the interests and objectives of government? Section I sketches the multiple tasks of ICs, in complex interdependence with other actors. Their core task is to adjudicate disputes through interpretation and application of international law by legal methods. This may also contribute indirectly to a range of further tasks. Section II addresses some aspects of the relation between normative legitimacy of ICs and descriptive legitimacy – actors’ beliefs therein. Section III shows how a wide range of legitimacy challenges concern ways ICs fail to carry out their tasks. This account does not seek to provide substantive arguments or seek to show that all such criticisms are correct. The aims are rather to make many such criticisms comprehensible as legitimacy concerns, to provide a rationale for popular taxonomies of legitimacy criticisms, and to indicate which premises and arguments are required for such criticisms to be sound.[D0I/LINK] [SSRN]