The Supreme Court, the highest court in the UK, was the Appellate Committee of the House of Lords until 2009. It is now housed in the neo-Gothic Middlesex Guildhall (1913) on Parliament Sq. There's a permanent exhibition looking at the work and history of the court as well as the building's history on the lower ground floor. There's a self-guided tour booklet (£1) or guided tours (adult/child £7/free) at 11am, 2pm and 3pm on Friday.
During the Supreme Court's summer recess (August to early September), there are also tours at 11am and 2pm on most weekdays. Members of the public are welcome to observe cases when the court is sitting (11am to 4pm Monday, 10.30am to 4pm Tuesday to Thursday). For who or what's on trial, ask for a list at reception, or go to the 'Current Cases' section of the Supreme Court website.