Back the Blue Act of 2017 This bill amends the federal criminal code: to establish new criminal offenses for killing, assaulting, and fleeing to avoid prosecution for killing a judge, law enforcement officer, or public safety officer; to expand the list of statutory aggravating factors in death penalty determinations to also include the killing or attempted killing of a law enforcement officer, judge, prosecutor, or firefighter or other first responder; and to broaden the authority of federal law enforcement officers to carry firearms. The bill amends the federal judicial code to limit federal court review of challenges to state court convictions for killing a public safety officer or judge. It amends the Revised Statutes of the United States to limit the recovery of compensatory damages and prohibit the recovery of attorney's fees by an individual for a violation of rights that occurs as a result of (or in the course of) conduct that constitutes a felony or crime of violence. Finally, it directs the Department of Justice to make grants to law enforcement agencies and nongovernmental organizations to promote trust and improve relations between law enforcement agencies and the communities they serve.