Robert Dundas was the eldest son of Robert Dundas of Arniston (1685–1753), Lord President of the Court of Session, by Elizabeth Watson, his first wife.
He was educated first at home and at school, and then at the University of Edinburgh.
In 1733, he proceeded to Utrecht University, then celebrated for the teaching of Roman law, and also visited Paris.

Returning to Scotland in 1737, Dundas was admitted an advocate in 1738.
He was quick, ingenious and eloquent, and had a retentive memory.
Like his father, he was convivial and shirked drudgery.
He is said, though a good scholar, never to have read through a book after leaving college.
Being solely ambitious of attaining to the bench, he refused many cases, especially those which involved writing papers, and took only such work as seemed to lead to advancement.
For his first five years, his fees only averaged £280 per annum.
Through the favour of the Carteret administration, he was appointed Solicitor General for Scotland on 11 August 1742, and, no change occurring in the Scotch department on Lord Wilmington's death, held that post through the arduous and responsible times of the Jacobite plots and the rising of 1745.
Being, however, unable to act easily with Lord Milton, the lord justice clerk, he resigned in 1746 upon the change of ministry, but was at once elected dean of the faculty.

On 16 August 1754, Dundas was appointed Lord Advocate, having been returned for Midlothian unopposed on 25 April at the general election.
While in parliament, he opposed the establishment of a militia in Scotland, and, as lord advocate, was largely occupied in settling the new conditions of the highlands, and in disposing of his great patronage so as to enhance the family influence.
But one speech of his in parliament is recorded,
Dundas was appointed a commissioner of fisheries on 17 June 1755, and on the death of Robert Craigie, he became lord president of the court of session, 14 June 1760.
He found upwards of two years' arrears of cases undecided, and having by great efforts disposed of them, he never allowed his case-list to fall into arrears again.
He was the best lord president who had filled the office, short but weighty in his judgements, thorough in his grasp of the cases, indignant at chicane, a punctilious guardian of the dignity of the court, a chief who called forth all the faculties of his colleagues.
Having, on 7 July 1767, given the casting vote against the claimant, Archibald Stewart, in the Douglas peerage case, he became very unpopular, and during the tumultuous rejoicings at Edinburgh, after the House of Lords had reversed that decision on 2 March 1769, the mob insulted him and attacked his house.
In his latter years his eyesight failed, and after a short illness he died at his house in Adam's Square on 13 December 1787, and was buried with great pomp at Borthwick on 18 December
Robert Dundas was the eldest son of Robert Dundas of Arniston (1685–1753), Lord President of the Court of Session, by Elizabeth Watson, his first wife. He was educated first at home and at school, and then at the University of Edinburgh. In 1733, he proceeded to Utrecht University, then celebrated for the teaching of Roman law, and also visited Paris.
Returning to Scotland in 1737, Dundas was admitted an advocate in 1738. He was quick, ingenious and eloquent, and had a retentive memory. Like his father, he was convivial and shirked drudgery. He is said, though a good scholar, never to have read through a book after leaving college. Being solely ambitious of attaining to the bench, he refused many cases, especially those which involved writing papers, and took only such work as seemed to lead to advancement. For his first five years, his fees only averaged £280 per annum. Through the favour of the Carteret administration, he was appointed Solicitor General for Scotland on 11 August 1742, and, no change occurring in the Scotch department on Lord Wilmington's death, held that post through the arduous and responsible times of the Jacobite plots and the rising of 1745. Being, however, unable to act easily with Lord Milton, the lord justice clerk, he resigned in 1746 upon the change of ministry, but was at once elected dean of the faculty.
On 16 August 1754, Dundas was appointed Lord Advocate, having been returned for Midlothian unopposed on 25 April at the general election. While in parliament, he opposed the establishment of a militia in Scotland, and, as lord advocate, was largely occupied in settling the new conditions of the highlands, and in disposing of his great patronage so as to enhance the family influence. But one speech of his in parliament is recorded,
Dundas was appointed a commissioner of fisheries on 17 June 1755, and on the death of Robert Craigie, he became lord president of the court of session, 14 June 1760. He found upwards of two years' arrears of cases undecided, and having by great efforts disposed of them, he never allowed his case-list to fall into arrears again. He was the best lord president who had filled the office, short but weighty in his judgements, thorough in his grasp of the cases, indignant at chicane, a punctilious guardian of the dignity of the court, a chief who called forth all the faculties of his colleagues. Having, on 7 July 1767, given the casting vote against the claimant, Archibald Stewart, in the Douglas peerage case, he became very unpopular, and during the tumultuous rejoicings at Edinburgh, after the House of Lords had reversed that decision on 2 March 1769, the mob insulted him and attacked his house. In his latter years his eyesight failed, and after a short illness he died at his house in Adam's Square on 13 December 1787, and was buried with great pomp at Borthwick on 18 December
Robert Dundas of Arniston, the younger