Son of Robert Dundas (1685-1753) and Elizabeth Watson, dau of Robert Watson of Muirhouse. Brother of Henry Dundas (1742-1811). Married (1741) to Henrietta Carmichael (d. 1755), dau of Sir James Carmichael, Bart of Bonnytoun. Married (1756) to Jean Grant, dau of William Grant, Lord Prestongrange. He had at least 10 children.
Dundas studied law at Utrecht University (as did David Dalrymple and James Boswell himself after him) and in 1738 was called to the bar. Solicitor General for Scotland (1742-1746). Member of Parliament of Midlothian (1754-?). Lord Advocate for Scotland (1754-1760). Lord President of the Court of Session (1760-1787). From 1754 until 1760 he was a Member of Parliament for the county of Edinburghshire and he held, for the remainder of his life, a large influence on Scottish politics.
Dundas came from a family of proud traditions within the legal profession, his father having also reached the high rank of Lord President (as the 3rd Lord Arniston), his grandfather having been a judge in the Court of Session and his great-grandfather (the 1st Lord) having been a senator.
Following his death in 1787, the poet Robert Burns penned the poem "On The Death of Lord President Dundas" in which he wrote:
"O heavy loss, thy country ill could bear!
A loss these evil days can ne'er repair!"
Boswell probably knew Dundas from a fairly early age, Dundas being an advocate and judge in Edinburgh and thus a colleague of Boswell's father. He mentions visiting Arniston House on October 28, 1762, together with John Webster, being "but dull at Arniston". We can only assume that Robert Dundas himself was also present, although Boswell does not mention him specifically.
On June 29, 1763, shortly before leaving for Utrecht, Boswell wrote to Dundas "that experience had now taught me that my father is as wise as myself, and that I am to follow his plan of life. I thank his Lordship for his former good offices while I was an idler, and hope he will not withhold them from me when I endeavour to be a man of business."
After an almost year-long lapse in his journal, Boswell began writing it again in January 1767. The earliest surviving entry in the resumed journal is for Saturday, January 10, when Boswell was again visiting Dundas at Arniston House, where he stayed until the 12th. Boswell seems to have spent quite some time in Dundas' library, and he also enjoyed the company of "Miss Dundas", presumably the Lord President's daughter Henrietta.