"Indian tribes are quasi-sovereign entities that enjoy all the sovereign powers that are not divested by Congress or inconsistent with the tribes' dependence on the United States. As a general rule, this means that Indian tribes cannot exercise criminal or civil jurisdiction over nonmembers. There are two exceptions to this rule for criminal jurisdiction. First, tribes may exercise criminal jurisdiction over nonmember Indians. Second, tribes may try non-Indians who commit dating and domestic violence crimes against Indians within the tribes' jurisdictions provided the non-Indians have sufficient ties to the tribes. There are three exceptions to this rule for civil jurisdiction. First, tribes may exercise jurisdiction over nonmembers who enter consensual relationships with the tribe or its members. Second, tribes may exercise jurisdiction over nonmembers within a reservation when the nonmember's conduct threatens or has some direct effect on the political integrity, the economic security, or the health or welfare of the tribe. These first two exceptions, enunciated in the case of Montana v. United States, are based on the tribes' inherent sovereignty, and exercises of jurisdiction under them must relate to a tribe's right to self-government. Third, Indian tribes may exercise jurisdiction over nonmembers when Congress authorizes them to do so. Congress may delegate federal authority to the tribes, or re-vest the tribes with inherent sovereign authority that they had lost previously. Indian tribes may also exercise jurisdiction over nonmembers under their power to exclude persons from tribal property."
CRS Report for Congress, R43324