Back the Blue Act of 2021 This bill establishes new criminal offenses for killing, assaulting, and fleeing to avoid prosecution for killing a judge, law enforcement officer, or public safety officer. Additionally, the bill expands 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; broadens the authority of federal law enforcement officers to carry firearms; and limits federal court review of challenges to state court convictions for killing a public safety officer or judge. 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.