Answer by taking a quote from the following article:

Leonard Peltier (born September 12, 1944) is a Native American activist, a citizen of the Anishinabe & Dakota/Lakota Nations, and member of the American Indian Movement (AIM). In 1977, he was convicted and sentenced to two consecutive terms of life imprisonment for first-degree murder in the shooting of two Federal Bureau of Investigation (FBI) agents during a 1975 conflict on the Pine Ridge Indian Reservation. Peltier's indictment and conviction have been the subject of much controversy; Amnesty International placed his case under the "Unfair Trials" category of its Annual Report: USA 2010. Peltier is incarcerated at the United States Penitentiary, Coleman in Florida.

Peltier has made a number of appeals against his murder convictions.  In 1986, Federal Appeals Judge Gerald W. Heaney, concluded, "When all is said and done ... a few simple but very important facts remain. The casing introduced into evidence had in fact been extracted from the Wichita AR-15." In his 1999 memoir, Peltier admitted that he fired at the agents, but denies that he fired the fatal shots that killed them.  A cartridge case from the Wichita AR-15 was found in the trunk of Agent Coler's car, and admitted as evidence at Peltier's trial in Fargo, North Dakota. Also admitted as evidence was the fact that no person involved in shooting at the agents, other than Peltier, possessed an AR-15 rifle.  The journalist Scott Anderson said that in a 1995 interview with Peltier, he sought answers to the contradictions he had found in Peltier's accounts of the incident on June 26, 1975. When asked about the guns he carried that day, Peltier listed a .30-30, a .303, a .306, a .250 and a .22, but he did not remember the AR-15.  The former United States Attorney General Ramsey Clark has served pro bono as one of Peltier's lawyers and has aided in filing a series of appeals on Peltier's behalf. In all appeals, the conviction and sentence have been affirmed by the 8th Circuit Court of Appeals. The last two appeals were Peltier v. Henman, 997 F. 2d 461 in July 1993 and United States v. Peltier, 446 F.3d 911 (8th Cir. 2006) (Peltier IV) in 2006.

What happened during the appeals process?
Peltier has made a number of appeals against his murder convictions.