Son of Alexander Hume (1675-1740), 2nd Earl of Marchmont, and Margaret Campbell (d. 1722) of Cessnock. Married (1731) to Anne Western of London (1708-1747). Married (1748) to Elizabeth Crompton. Member of Parliament for the Country of Berwick (1734-1740) as Lord Polwarth, and sometime leader of the opposition. Appointed First Lord of Police in 1747. Keeper of the Great Seal of Scotland (1764-1794). A friend of Alexander Pope.
Boswell visited Marchmont House with Henry Home on October 18, 1762. He met his lordship, describing him as "a man of fire and good elocution, by which he was distinguished as a commoner, and by having more knowledge and application to business than is commonly found amongst the nobility, he is now of consequence in the House of Lords. He is a true politician, so that the qualities of the heart must be dispensed with. He is artful and false; he is insolent and a coward. This last division of his character I gathered from Lord Kames. But as they are inveterate enemies, it is possibly too deeply coloured. He behaved to me with much affability of address".
In 1831 was published "A Selection from the Papers of the Earls of Marchmont; Illustrative of Events from 1685 to 1750". Some correspondence of the 3rd Earl is included. The book is available in original as well as print-on-demand editions via AbeBooks.