Opponents are fighting to stop Theresa May triggering Article 50 of the Lisbon Treaty to leave the European Union without the prior authorisation of Parliament.
Government lawyers will argue before three judges at the High Court in London that the Prime Minister is legally entitled to use the royal prerogative to start the Brexit process.
Mrs May announced at the Conservative Party conference that she intends doing so by the end of March.
Campaigners say only Parliament is empowered to authorise service of the Article 50 notice and Mrs May’s stance threatens to undermine its sovereignty and the rule of law.
The lead case is being brought by Gina Miller, an investment fund manager and philanthropist living in London who voted Remain in the EU referendum on June 23.
Ms Miller said when given permission to launch her action: “This case is all about the sovereignty of Parliament. It is very important that the (Article 50) issues are dealt with in a serious and grown-up way.
“We are making sure that happens.”
The first applicant to lodge a Brexit legal challenge was London hairdresser Deir Dos Santos, and he is among a wide group of “concerned citizens” supporting Ms Miller’s application for judicial review.
Their lawyers contend it is a fundamental constitutional requirement that there should be full scrutiny by both houses of Parliament, followed by new legislation.
Notification of Article 50 by royal prerogative, they argue, would hit rights established by the European Communities Act 1972, which made EU law part of UK law. But only Parliament has the sovereign power to repeal the 1972 Act.
The case is being heard by Lord Thomas, the Lord Chief Justice, sitting with Sir Terence Etherton, Master of the Rolls, and Lord Justice Sales.
Attorney General Jeremy Wright, the Government’s leading law officer, will argue in court that the challenge lacks legal merit.
He said: “The country voted to leave the EU in a referendum approved by Act of Parliament.
“There must be no attempts to remain inside the EU, no attempts to rejoin it through the back door, and no second referendum.
“The result should be respected and the Government intends to do just that.”
Other applicants wishing to intervene include Britons living in France campaigning as Fair Deal for Expats and individuals from other countries living in the UK anxious to secure guarantees for rights gained as EU citizens.
A group referred to as “the AB Parties” say they are representative of a large class of “ordinary, poor or otherwise vulnerable” people and their children whose fundamental human rights and stability are at stake.
Because of the urgency and constitutional importance of the case, any appeal is expected to be heard by the Supreme Court, the highest court in the land, before the end of the year.
DOWNLOAD THE SCOTSMAN APP ON ITUNES OR GOOGLE PLAY