Robert Weston was the seventh son of John Weston (born c.1470, died c.1550), a tradesman of Lichfield, Staffordshire. The Weston family of Gloucestershire, who produced another senior Irish judge in William Weston were probably cousins. He entered All Souls, Oxford and was elected Fellow in 1536. He studied Civil Law, and attained the degree of BCL on 17 February 1538 and DCL on 20 July 1556. From 1546 to 1549 he was a principal of Broadgate Hall, and at the same time deputy reader in civil law to the University, under Dr John Story.
On 12 January 1559 Weston was created Dean of the Arches and was a commissioner for administering the oaths required of ecclesiastics under to the Act of Uniformity. He was consulted in regard to the Queen's Commission issued on 6 December 1559 for confirming Matthew Parker as Archbishop of Canterbury and was included in a commission issued on 8 November 1564 to inquire into complaints of piratical depredations committed at sea on the subjects of the King of Spain.
The Lord Deputy of Ireland Sir Henry Sidney, requested that Weston be nominated for the post of Lord Chancellor of Ireland in succession to Hugh Curwen Archbishop of Dublin in April 1566. After a year, on 10 June 1567 Queen Elizabeth told Sidney that after good deliberation she had made the "choice for the supply of room of Chancellor by naming thereunnto our trusty well-beloved Doctor Weston, dean of the arches here, a man for his learning and approved integrity thoroughly qualified to receive and possess the same" and "that for some increase of his living whilst he remaineth in our service there she was pleased to give unto him the Deanery of St Patrick's whereof the Bishop of Armagh is now dean and yet to leave it at our order, as we know he will". Weston arrived in Dublin early in August and was sworn into office on 8 August 1567. He and Sir William Fitzwilliam the Vice-Treasurer of Ireland were sworn Lord Justices in Christ Church Cathedral, Dublin on 14 October. Weston addressed the Irish Parliament when it was summoned on 17 January 1568.
Weston was described as follows
"A man in his time most godlie, upright, and virtuous, and such a one as that place was not possessed of the like in many currents of years. In his life he was most virtuous and godlie; in matters of council most sound and perfect; in justice most upright and uncorrupted in hospitalitie very bountie and liberal; in manners and conversation most courteous and gentle; faithful to his Prince, firm to his friend, and courteous to all men; and as was his life, so was his death, who a little time before the same called his household, and gave them such godlie instructions as to their callings appertained; then he set his private things in order, and he spent all the time that he had in praiers and exhortations"
Weston married (1st) Alice Jenyngs, daughter of Richard Jenyngs of Barr, near Lichfield. They had a son John, and three daughters, of whom we know most of Alice, who married firstly Hugh Brady, Bishop of Meath, by whom she had numerous children including Luke and Nicholas, and secondly Sir Geoffrey Fenton, by whom she was the mother of Catherine, who married Richard Boyle, 1st Earl of Cork and of Sir William Fenton. He married (2nd) Alice Bigges, daughter of John Bigges of Isleworth and widow of G. Aunsham of Heston. They had no issue. Through his daughter Alice Weston was the ancestor of another Lord Chancellor of Ireland, Maziere Brady.
- Chapters of Dublin Chapter XXVIII. Life of Lord Chancellor Weston
- Browne Willis Notitia parliamentaria, or, An history of the counties, cities, and boroughs in England and Wales: ... The whole extracted from mss. and printed evidences 1750 p.66
- Calendar State Papers Dom 1547-80, p246
- Holinshed's Chronicle, vol. vi. p. 373.