Back in 1978, the Supreme Court ruled tribes can’t prosecute non-Indians for crimes committed on reservations, but also said Congress could give tribes such authority. (Oliphant v. Suquamish Tribe)
Congress did exactly that in the Violence Against Women Reauthorization Act of 2013, which gives tribes criminal jurisdiction over domestic or dating violence occurring in Indian Country regardless of the perpetrator’s Indian or non-Indian status.
Generally speaking, the federal government prosecutes non-Indians who commit crimes on tribal reservations, which are considered federal territory, and also prosecutes Indians for on-reservation “major crimes” (murder, rape, etc.) designated in the Major Crimes Act, but otherwise Congress has delegated authority over crimes committed by Indians on Indian lands to tribal governments.
Until now, the 1978 ruling has meant that a non-Indian husband who beats up an Indian wife in their home on a reservation couldn’t be prosecuted in tribal courts. The VAWRA changes that, generally taking effect in March 2015, but under a provision authorizing pilot projects, the Attorney General has authorized three tribes to begin exercising such jurisdiction on February 20, 2014.
The three tribes are the Tulalips in Washington, the Umatillas in Oregon, and the Pascua Yaquis in Arizona.