Her death, just weeks before the presidential election, offers Republicans a chance to lock in a conservative majority for decades to come, on a court where justices are appointed for life.
The stakes are high as the decision could affect such weighty issues as abortion, healthcare, gun control and gay rights.
They are pushed even higher in a bitter election year when the justices can play a decisive role in legal wrangling over a contested result — such as when they ruled in George W. Bush's favor to end the 2000 election debacle.
Trump has already named two justices during his term as president, Brett Kavanaugh and Neil Gorsuch, giving conservatives a 5-4 majority before Ginsburg's death, though that does not guarantee rulings in Trump's favor — there have been several recent examples of conservatives siding with their progressive colleagues.
Trump, who is lagging in the polls behind Democratic opponent Joe Biden, has another powerful incentive to move ahead: providing a jolt of enthusiasm among his anti-abortion and evangelical supporters.
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But, with 45 days to go before the election and early voting already begun in some states, galvanized Democrats are pushing back furiously.
Biden said Friday that "the voters should pick the president and the president should pick the justice for the Senate to consider".
'Nothing is off the table'
While Democrats' options seem limited, Senate minority leader Chuck Schumer told party members Saturday that if Republicans press ahead, then "nothing is off the table", according to media reports.