With the recent addition of Justice Brett Kavanaugh, President Donald Trump's embattled Supreme Court pick who faced sexual assault allegations during his nomination process, they are the minority on the court.
Just two days after the mid-term elections, the significance of Republicans maintaining control of the Senate - which confirms presidential nominees - has been put in stark relief.
Ginsburg joined the court in 1993.
"I pray for Ruth Bader Ginsberg every day", Democratic Senator Bob Menendez said Sunday.
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She was appointed to the U.S. Court of Appeals in 1980 by President Jimmy Carter for the District of Columbia. Fans of Notorious RBG, as she is sometimes known, immediately launched into grief - and action.