The UK Government has urged the country's highest court to overturn a ruling that the Prime Minister must seek MPs' approval to trigger the process of taking Britain out of the European Union.
Eleven Supreme Court justices - a record to hear an appeal - are hearing the challenge over the country's Brexit strategy, which has attracted world-wide attention from the media and public.
The Scottish and Welsh governments and the Attorney General for Northern Ireland are all intervening in the Supreme Court case.
A ruling is not expected to be delivered until the new year.
The hearing will be the most televised UK case ever, with the proceedings streamed on the Supreme Court website and broadcast on television.
In a decision on November 3 that infuriated Brexiteers, three High Court judges said Theresa May lacked power to use the royal prerogative to trigger Article 50 of the Lisbon Treaty and start the two-year process of negotiating Brexit without the prior authority of Parliament.
Now the justices are to have their say regarding one of the most important constitutional cases in British legal history.
The Government's top law officer, Attorney General Jeremy Wright, told the justices that the case was of "great constitutional significance in which there is understandable and legitimate interest both inside and outside this courtroom".
"Secondly, in the light of what followed the Divisional Court (High Court) judgment, it should be said with clarity this is a case which the claimants brought perfectly properly and which it is now perfectly proper for this court to decide."
Mr Wright said the High Court had reached the "wrong" decision. It was for the Government to exercise prerogative powers in the conduct of the UK's affairs on the international plane.
He told the judges that triggering Article 50 "will not be an exercise of the prerogative right on a whim or out of the blue", but was part of a process in which "Parliament has been fully and consciously involved".
Mr Wright said the use of the prerogative in the circumstances would be lawful. The prerogative was not "an ancient relic", but a "constitutional necessity".
He said the legislation enabling the EU referendum had been passed with the "clear expectation" that the Government would implement the result, and that Parliament had had the opportunity to restrict the Government's power to trigger Article 50, but had chosen not to do so.
He said: "If this is all about standing up for Parliament, I say Parliament can stand up for itself."
James Eadie QC, also representing the Government, described the prerogative as "a long-standing, well-recognised set of powers firmly established in our constitutional arrangements" which were "fundamental to our constitution and essential to effective government".
He submitted: "The Government has legal power to give notice pursuant to Article 50."
If the appeal is unsuccessful, and any potential further appeal to the European Court of Justice in Luxembourg also fails, the Government's plans for Brexit could be thrown into disarray.
But Mrs May has made it clear she still intends to give an Article 50 notification by the end of next March to start the leave negotiations with 27 other EU countries.
At the outset of the hearing Lord Neuberger, the court's president, said all parties had been asked whether they wished any of the judges to stand down.
He said that all parties to the appeal had stated that they have no objection to any of the justices sitting on the appeal.
The announcement follows media reports and comments that have questioned the independence of members of the judiciary.
Lord Neuberger also said individual members of the public in the case had received "threats of serious violence and unpleasant abuse in emails and other electronic communications" and warned those responsible that "legal powers" existed to deal with them.
He said: "Threatening and abusing people because they are exercising their fundamental right to go to court undermines the rule of law.
"Anyone who communicates such threats or abuse should be aware that there are legal powers designed to ensure that access to the courts is available to everybody."
Lord Neuberger is leading a panel including Lady Hale, Lord Mance, Lord Kerr, Lord Clarke, Lord Wilson, Lord Sumption, Lord Reed, Lord Carnwath, Lord Hughes and Lord Hodge.
In his opening remarks Lord Neuberger stressed the court was aware of the "strong feelings associated with the many wider political questions surrounding the United Kingdom's departure from the European Union" - but those questions were not the subject of the appeal.
He said: "This appeal is concerned with legal issues and, as judges, our duty is to consider those issues impartially, and to decide the case according to the law. That is what we shall do."
Brexit Secretary David Davis is leading the Government's historic legal action. His team of lawyers, headed by Mr Wright, will argue in the four-day hearing that the three High Court judges erred over Article 50 and its use was legally justified by the June 23 referendum vote in favour of quitting the EU.
Lord Thomas, the Lord Chief Justice, gave the ruling blocking the use of Article 50. Two other top judges - Master of the Rolls Sir Terence Etherton and Lord Justice Sales - agreed.
Even though it was emphasised to a packed court that they were deciding "a pure question of law" and not expressing any view about the merits of leaving the European Union, they faced fierce criticism from Leave campaigners and an accusation that they were "enemies of the people".
The High Court ruling was won by Gina Miller, 51, an investment fund manager and philanthropist who was selected to bring the lead case.
She reported that her high-profile role had led to death threats and she had spent £60,000 on security, but she is returning to the battle represented once more by Lord Pannick QC.
Her case is being supported by "concerned citizens" drawn from all walks of life, including London hairdresser Deir Dos Santos, 37, who helped start the legal battle over Brexit but, say his lawyers, has been forced underground after receiving "vile" hate mail.