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Ginsburg died on Friday of metastatic pancreatic cancer at her home in Washington, DC, surrounded by her family, the statement said.
Earlier this year, Ginsburg said she was undergoing chemotherapy for a recurrence of cancer.
She was a prominent feminist who became a figurehead for liberals in the US.
Ginsburg was the oldest sitting justice on the Supreme Court, having served 27 years on the nation's highest court.
"Our Nation has lost a jurist of historic stature," Chief Justice John Roberts said in a statement on Friday. "We at the Supreme Court have lost a cherished colleague. Today we mourn, but with confidence that future generations will remember Ruth Bader Ginsburg as we knew her - a tireless and resolute champion of justice."
US Supreme Court justices serve for life or until they choose to retire, and supporters had expressed concern that a more conservative justice could succeed Ginsburg.
Ginsberg's death will spark a political battle over who will succeed her, spurring debate about the future of the Supreme Court ahead of November's presidential election.