Justice Sotomayor argues that the private intent of Trump's official acts could be inferred from evidence presented in court.

But, she added, "you wouldn't expunge that as evidence" and you would tell the jury not to consider some of these actions during their deliberations.

"I'm not sure that I understand why your problems couldn't be taken care of at trial with an instruction, if the Court were to find - I'm not even sure how they could, but if it were to find - that some public acts could not be the basis of criminal liability," she tells Sauer.

Trump's lawyer replies that the reason his client is calling for blanket immunity is because "it would be difficult for some of these prosecutions to proceed" in their entirety.

Justice Elena Kagan just raised the OLC - the Office of Legal Counsel.

So what is it?

The OLC is an office in the Justice Department, one that assists the US attorney general - the country's top lawyer - in giving legal advice to the president.

The US Supreme Court doesn't allow cameras in the courtroom when it's in session, but you can listen to proceedings by clicking the Play button at the top of this page.

We'll also have live text updates of the opening arguments right here. Stay tuned.

Image source, Bloomberg

Justice Amy Coney Barrett - a conservative Trump appointee - chimes in for the first time, asking Sauer a series of quickfire questions about "private" versus "public" acts - a distinction that many of the justices have touched on this morning.

She asks if Sauer agrees that a president can be prosecuted for private, non-official, acts. Sauer agrees.

Barrett then leads Sauer through a number of allegations in Trump's indictment, like meeting with an attorney willing to spread false election claims.

Sauer disputes the charges, but agrees those acts would be "private".

We're now hearing about arguments at the Appeals Court, when a judge brought up the hypothetical scenario of Seal Team Six, an elite military unit, being ordered to assassinate someone.

Tump’s lawyer has repeatedly argued that a president could be protected from prosecution for doing so.

"I don't want to slander Seal Team Six because they're - no seriously - they're honourable," Justice Alito says.

"They're honourable officers and they are bound by the Uniform Code of Military Justice not to obey unlawful orders."

But he adds that someone could say it's not plausible that action would be legal.

"I'm sure you've thought of lots of hypotheticals where a president could say 'I'm using an official power' and he uses it in an outrageous manner."

Anthony Zurcher
BBC North America correspondent

Donald Trump’s lawyer, John Sauer, isn’t getting a particularly warm reception from the Supreme Court justices as he tries to make the case that former presidents have sweeping immunity from prosecution for actions taken while in office.

Right out of the gate, Chief Justice John Roberts questioned whether a president offering an ambassadorship in exchange for a bribe would be an official act that is immune from prosecution.

The chief justice – who could be a key vote on this court - seemed dissatisfied with Saeur’s response.

And he wasn’t the only one who pressed Trump’s lawyer on what constitutes an official act and how to draw the line between private actions and those done as part of presidential duties.

“My question is whether the very robust form of immunity you’re advocating is necessary,” asked Samuel Alito, one of the court’s most conservative justices.

Five of the nine Supreme Court justices will have to agree to form a majority opinion. At least in the early stages of these oral arguments, few justices have seriously entertained the notion that Trump is totally protected from prosecution.

Even if the court doesn’t fully side with Trump’s team, however, they could issue a ruling that further delays Trump’s criminal trials – perhaps pushing it beyond November’s presidential election.

Justice Samuel Alito questions how one can determine whether the actions of Donald Trump or any other president were "official conduct" or "private conduct".

He notes that it seems that would likely be decided on "purely objective grounds".

Given that may be the case, Alito lets Sauer know that he is sceptical whether the "very robust form of immunity that you're advocating is really necessary".

Image source, Bloomberg

Justice Ketanji Brown Jackson, a liberal justice and the most recent addition to the bench, says the crucial decision before the court is what is an "official act".

She asks: When the president is using the trappings of his office to achieve a personal gain is he acting officially?

Sauer responds that that opens up questions about the president's personal motives for every action they take.

Justice Jackson responds that "every" president has understood that there's a threat of prosecution and "they've continued to function".

If presidents had total immunity, she asks, then "what was Nixon's pardon about?" referring to the presidential pardon of Richard Nixon.

The US Supreme Court doesn't allow cameras in the courtroom when it's in session, but you can listen to proceedings by clicking the Play button at the top of this page.

We'll also have live text updates of the opening arguments right here. Stay tuned.

Image source, The Washington Post

Justice Sonia Sotomayor jumps in to contrast the controversial actions of former presidents mentioned by Sauer with the one Donald Trump is accused of.

Barack Obama's drone strike was taken as a "reasonable" measure to protect against terrorists, she says.

But Trump's actions at question in this case appear to be ones he took "for personal reasons".

Sotomayor asserts that Sauer is arguing "even if you did it for personal gain, we won't hold you responsible".

By that logic, she continues, "a president is entitled not to make a mistake but more than that - a president is entitled, for total personal gain, to use the trappings of his office without facing criminal liability".

Image source, Reuters

Chief Justice John Roberts asks if a president would have immunity if they appointed an ambassador in exchange for a bribe.

Roberts points out that a bribe wouldn’t appear to be an official act.

He adds that if you're prosecuted because you accepted a million dollars, are you "not supposed to say what it's for because the 'what's for' part is within the president's official duties?"

"There has to be an independent source of evidence for that," Trumps lawyer, Dean John Sauer replies.

Sauer says any "official act" that an executive does as president would be protected by presidential immunity.

The first question of the day comes from Justice Clarence Thomas, a conservative, and the longest-serving member of the court.

Thomas asks Trump's lawyer Dean John Sauer where "precisely" presidential immunity comes from, and what an "official act is" - two crucial questions in this case.

Dean John Sauer, Donald Trump's lawyer, is kicking off this session.

"Without presidential immunity from criminal prosecution, there can be no presidency as we know it," he begins.

Sauer, who also argued this case before the DC Appeals Court that ruled unanimously against Trump earlier this year, notes again that no president has been charged and criminally prosecuted before.

Doing so to Trump will open future presidents to "a looming threat that will distort the presidency" and open them up to "de facto blackmail and extortion from his political rivals".

Presidents should know they can have the ability to boldly move forward with their most controversial decisions.

If not, he contends, George W Bush could be indicted for lying to Congress to induce war in Iraq, or Barack Obama could face prosecution for killing US citizens abroad with drone strikes. Even the incumbent, Joe Biden, could face legal action for "unlawfully inducing immigrants to enter the country illegally", he argues.

The gavel has been sounded as the nine robed justices of the Supreme Court enter the courtroom and take their respective seats.

Court is now in session.

You can follow the oral argument live by pressing the Play button at the top of this page - and stay tuned as we break it all down for you.

Image source, Getty Images

We're expecting Justice Clarence Thomas to ask the first question today.

But some critics of the conservative-dominated Supreme Court have called for him to step away from this case, over ties to pro-Trump activists who sought to overturn the 2020 election result.

Thomas’s wife, Virginia “Ginni” Thomas, had been active in the legal effort to block Biden’s victory, and attended the rally on 6 January that preceded the riot at Congress.

She also sent text messages, fanning conspiracies about the 2020 election, to top White House and Republican state officials, in an effort to overturn Trump’s loss, according to documents obtained by the congressional committee that investigated the Capitol riot.

In 2022, Ginni told the committee: "I regret all of these texts."

Justice Thomas, who joined the court in 1991, has ignored the calls to step aside.

Image source, Getty Images

Donald Trump’s legal team are citing a 1982 case, Nixon vs Fitzgerald, as their main argument.

Arthur Fitzgerald was an analyst with the air force department and in 1968 he became a whistleblower, testifying in Congress about budget blowouts and technical problems.

When he was fired two years later, he claimed it was in retaliation for his testimony.

Later, Fitzgerald sued several White House officials including Richard Nixon, who was president at the time he was fired.

Two lower courts rejected Nixon’s plea for immunity, but the Supreme Court went the other way and ultimately granted Nixon presidential immunity from civil damages - basically, he couldn’t be sued for acts committed while in office.

But this case is different to Donald Trump’s – Nixon vs Fitzgerald was a civil case, while Trump’s is criminal.

Consequences of the case will reverberate for years to comepublished at 14:51 British Summer Time 25 April

14:51 BST 25 April

Gary O'Donoghue
Washington correspondent, outside the Supreme Court

Image source, Getty Images

The Supreme Court has never heard a case like this, mainly because no former president in the history of the republic has been charged with crimes allegedly committed in office.

The detailed constitutional arguments will be scrutinised in the coming hours, but the consequences of this case will reverberate for years to come.

If the former president persuades the nine justices he has total immunity, then the criminal cases Trump is facing will unquestionably fall away.

Granting total immunity may be unlikely, but the delays are making it increasingly likely that several of these trials may not happen before the election.

Ironically, the case considered his least serious - the hush-money trial currently under way in New York - now takes on a much more significant aspect; since it, and any appeals, are likely to conclude before November. If Trump were to be convicted, he would go to the electorate as a felon - something the polls tell us is not very appealing to swing voters in key states.

Image source, Getty Images

Donald Trump has just spoken ahead of the Supreme Court hearing, saying the court "has a very important argument before it today".

But he won't be there to hear it. The former president is in New York, where he is on trial for felony charges related to an alleged hush money payment.

"I should be there," Trump says, adding "we don't want a ceremonial president, we want a real president."

We also have live coverage of his Manhattan trial - you can follow that here.

Anthony Zurcher
BBC North America correspondent

Donald Trump’s lawyers have fought over every inch of legal ground in the criminal cases against him, contesting judicial decisions and appealing them to higher courts whenever possible.

The unstated goal has been to delay a moment of final legal reckoning for as long as possible.

Even if the Supreme Court ultimately rules against Trump, the justices may have given him enough time to defeat the two federal cases that have been brought against him.

If he wins re-election in November, Trump could appoint Justice Department officials who are willing to drop the prosecutions – or, at the very least, put the cases on hold for his four-year presidential term.

If that doesn’t work, he could take the unprecedented step of pardoning himself for the alleged crimes.

The chance that such a delay strategy could succeed, regardless of the merits of the cases against him, is one of the reasons why some of Trump’s fiercest critics are angry that it took years for federal charges to be filed against him.

Read Anthony's full story on today's hearing here.

Image source, EPA

A veteran prosecutor, Jack Smith has spent two decades pursuing officials in the US and abroad, and is currently overseeing two separate criminal investigations into Trump.

The New Yorker started as an assistant district attorney in Manhattan in 1994, and climbed up the ranks there, pursuing violent gangs, white-collar fraudsters and public corruption cases.

He once spent a weekend sleeping in the hallway of an apartment building so he could convince a woman to take the witness stand in a domestic violence case, the Associated Press (AP) reported.

Now, as the US Justice Department’s special counsel, he has charged Trump over alleged efforts to overturn the 2020 election. Smith has also indicted the ex-president on 40 felony counts over his alleged mishandling of classified documents.

Trump has characterised Smith as a "deranged" man at the forefront of a "political witch hunt" against him.

• Read more on Jack Smith here.