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President Trump has said he will name a replacement to Ruth Bader Ginsburg by week's end and urged the Republican-controlled Senate to confirm his Supreme Court choice before 3 November.
The plan has launched a high-stakes battle ahead of the election.
Mr Trump would replace Ginsburg, a liberal stalwart who died on Friday aged 87, with a conservative.
The president appears to have secured enough support in the US Senate to win approval for his nominee.
This would cement a right-leaning majority on the court for decades.
The ideological balance of the nine-member court is crucial to its rulings on the most important issues in US law.
What happens next?
On Monday Mr Trump said that he was "constitutionally obligated" to nominate someone for the Supreme Court.
"We're looking at five incredible jurists... women that are extraordinary in every way. I mean, honestly, it could be anyone of them, and we're going to be announcing it on Friday or Saturday," he told supported at a rally in Ohio .
The president earlier had a private meeting at the White House with a potential nominee: Amy Coney Barrett, an appeals court judge who is backed by anti-abortion conservatives.
Once the president names a nominee, it is the Senate's job to vote on whether to confirm them. The Judiciary Committee will review the pick first, and then vote to send the nominee to the floor for a full vote.
Senate majority leader Mitch McConnell has vowed to hold a confirmation vote before the election in November. Democrats have accused him of hypocrisy.
Following the death of conservative justice Anthony Scalia in 2016, Mr McConnell refused to hold a vote to confirm a nominee put forward by then-President Barack Obama, a Democrat.
Image copyright Getty Images Image caption Mitch McConnell has promised a Senate vote on the nomineeMr Obama had nominated Merrick Garland in February of that year - months before the election - but Mr McConnell argued that Supreme Court justices should not be approved in an election year.
In 2017, Mr McConnell also changed Senate rules to allow for a simple majority (51 votes) to confirm nominees.
However, this time around, with a president of the same party, the Senate leader says because the Senate and White House are both Republican-held, unlike 2016, the nomination should proceed.
Do Senate Republicans have the votes?
The president's plan to appoint a justice was boosted on Monday after two closely watched senators of his party, Cory Gardner of Colorado and Charles Grassley of Iowa, signalled they backed moving ahead.
Their support may grant Republicans the 50 votes they need to confirm a justice, given that Vice-President Mike Pence can cast a tie-breaking vote if needed.
Republicans hold a 53-47 majority in the upper chamber.
Lindsey Graham, the powerful Republican chairman of the Senate Judiciary committee, said on Monday he would be "leading the charge to make sure that President Trump's nominee has a hearing, [and] goes to the floor of the United States Senate for a vote".
Mitt Romney, of Utah, remains undecided. Susan Collins of Maine and Lisa Murkowski of Alaska, have backed a delay in the vote.
Ms Collins said she had "no objection" to the process of reviewing a candidate beginning now, but that she did not believe the Senate should vote on the candidate before November's election. Ms Collins is facing a tough re-election bid this year.
Ms Murkowski said she "did not support taking up a nomination eight months before the 2016 election" and believed the "same standard must apply" now.
Even if Republicans lose their Senate majority on 3 November, the new Congress does not take office until 3 January, which would give them time to confirm Mr Trump's pick.
If the nominee is not confirmed by 20 January, Inauguration Day, they would have to be re-nominated by the president (whoever that ends up being).
Battle over Supreme Court
Image copyright Getty ImagesHow long does it take to confirm a Supreme Court judge?
Typically, it's a months long process to go from selection to confirmation - but there are no rules regarding this time frame.
Since 1975, it has taken about 70 days on average. This time, the election is just weeks away.
The last time lawmakers completed a confirmation this speedily was for Ginsburg's own selection in 1993. She was approved in 42 days.
What's at stake?
The highest court in the US is often the final word on highly contentious laws, disputes between states and the federal government, and final appeals to stay executions.
In recent years, the court has expanded gay marriage to all 50 states, allowed for President Trump's travel ban to be put in place, and delayed a US plan to cut carbon emissions while appeals went forward.
The court also handles reproductive rights issues like abortion - a highly contentious election issue, especially for one of Mr Trump's key Republican constituencies. Opponents of abortion have called for overturning abortion protections, and appointing judges who sympathise with this view is one of Mr Trump's pitches for re-election.
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Clara Spera, Ginsburg's grand-daughter, disclosed that the late justice's dying wish was not to be replaced until after the election. "She was concerned about this country and about the court that she served so diligently for over 27 years," Ms Spera told BBC World Service.
"I think that she would be heartened to know that there are many, many people who believe that we need to return to order and norms, and agree and want to fulfil that most fervent wish of hers," she said.
Ginsburg will lie in state at the US Capitol on Friday.
What's the reaction?
Speaker of the House Nancy Pelosi, the most powerful Democrat in Washington, suggested she may try to influence what happens next with the confirmation.
Mrs Pelosi told the New York Times she had "arrows in [her] quiver, in the House quiver" but would not offer more details.
On Sunday, Democratic presidential candidate Joe Biden said Mr Trump had "made clear this is about power, pure and simple".
He urged Senate Republicans to "please follow your conscience, let the people speak, [and] cool the flames that have been engulfing our country".
Image copyright ReutersDuring a vigil for Ginsburg on Sunday night, Democratic Senator Elizabeth Warren drew cheers from the crowd for criticising Republicans over the nomination.
"Mitch McConnell and his henchmen believe that they can ram through a Supreme Court justice only 45 days from Election Day," she said. "What Mitch McConnell does not understand is that this fight has just begun."
