The court's 11 justices will hear an appeal by Theresa May's government against the recent High Court ruling that only the Westminster Parliament has the authority to trigger article 50.
The hearing, which is being streamed live and saw people queuing to get in from early morning, is expected to last four days, with the verdict expected in January.
The case is about whether the law means that the UK government needs the authority of Parliament to trigger the process for the UK to leave the EU.
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The government argues it can start the article 50 process using "prerogative powers", a remnant of the era of all-powerful kings and queens.
Justices will be ruling on who has the legal power to change the rights of British citizens
The outcome will have implications for Theresa May's strategy for EU exit, but it is not a court case on whether or not Brexit actually takes place.
The UK voted to leave the EU, by a margin of 51.9 per cent to 48.1 per cent, in a referendum in June.
May has said she intends to officially notify the rest of the EU of the UK's intention to leave - beginning two years of talks over the terms of separation - by the end of March.
Meanwhile, UK Labour leader Jeremy Corbyn, speaking at the Party of European Socialists Council in Prague at the weekend, has repeated his assertion that Labour will not try to overcome the result of the EU referendum.
Corbyn said, "When it comes to Britain's referendum vote to leave the European Union we in the Labour party respect that decision, and we want to work together with Socialist and progressive parties across Europe to find the best possible solution that benefits both Britain and the EU in the Brexit negotiations.
"Labour is calling on the British government to guarantee the rights of all EU citizens before article 50 negotiations begin, and not to use them as a bargaining chip in negotiations."
He added, "Labour is pushing for Brexit negotiations to be carried out in a transparent manner, in a spirit that aims to find a deal that works for all across our continent."
Corbyn has invited leaders from Socialist parties across Europe to a special conference in London in February to debate Brexit-related issues.
Elsewhere, business leaders in Brussels are campaigning for English to become an official administrative language in the region, in a bid to attract companies from the UK after Brexit.
Jan De Brabanter, Secretary General of the Brussels Chamber of Commerce, said the idea had support from local businesses, and that many politicians in the Brussels region "can see the benefit."
However, he admitted that it would be "very hard" to achieve, as all three regions and the federal government would need to agree.