Scott was born in 1954, the son of John Scott, Earl of Dalkeith and his wife, Jane née McNeill, and was baptised with Princess Margaret as one of his godparents. He later attended Eton, was Page of Honour to the Queen Mother from 1967 to 1969. In 1973 his father inherited the Dukedoms of Buccleuch and Queensberry, and Scott took the courtesy title of Earl of Dalkeith. He later graduated from Christ Church, Oxford in 1976 with a Bachelor of Arts.

In 1981 he married Lady Elizabeth Marian Frances Kerr, a daughter of the 12th Marquess of Lothian (and a sister of Michael Ancram), and they later had four children:

Lord Dalkeith had a brief spell on the board of Border Television from 1989 to 1990, and in 1994 he joined the Millennium Commission as the representative for Northern England. Appointed a KBE in 2000 for his services to the Millennium Celebrations, Dalkeith left the commission in 2003. He is President of the National Trust for Scotland and is a Fellow of the Royal Society of Edinburgh. Dalkeith also served as Deputy Chairman of the (since abolished) Independent Television Commission, as a member of Scottish Heritage, on the board of the Winston Churchill Memorial Trust and was President of the Royal Scottish Geographical Society from 1999 to 2005.

The Duke currently lives in the Dabton area of Thornhill, Dumfries and Galloway near his father's home, Drumlanrig Castle.

On September 4, 2007 he inherited the titles Duke of Buccleuch and Duke of Queensberry upon his father's death.