All 11 justices sat together for the first time to hear the submissions from the UK government, lead respondent Gina Miller and other interested parties.
They must now decide if the government can trigger article 50 and begin the process of exiting the European Union without a vote in parliament.
Judgment is expected in January 2017.
Arthur Simpson-Kent, the partner of former EastEnders actress Sian Blake, appeals his whole life sentence for murdering her and their two children after learning she was going to leave him and take them with her. Watch the broadcast, in association with Sky News.
32-month prison sentence appealed: Joseph Smith was jailed in February for causing death by dangerous driving.
Appeal against sentence: Azhar Islam Khan, for tax evasion. Criminal defence solicitor jailed for 18 months in January for lying about income from his Wembley-based City Law Solicitors in order to steal more than £60,000.
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