Sir Robert Gunning, 1st Baronet GCB (8 June 1731 – 22 September 1816) was a British diplomat. He served as the British minister in Denmark 1765–1771, in Prussia in 1771 and in Russia 1772–1776. Gunning was appointed a Knight Companion of the Bath on the 9 July 1773 and the first Baronet of Eltham in the County of Kent on 27 October 1778.
The Gunnings were an Irish family. His country seat was at Horton which he purchased 1782; he was the eldest son of Robert Gunning - and his mother was Catherine, the daughter of John Edwards. The family was descended from Richard Gunning, who was an uncle of Peter Gunning, the Bishop of Ely who had settled in Ireland in the time of James I. The Gunning family stayed at Horton Hall until 1888 when it was sold to Pickering Phipps, the Northampton brewer. Later still, it was sold to George Winterbottom.
Gunning entered the diplomatic service, and on 23 November 1765 was appointed minister resident at the court of Denmark, where he arrived in April of the following year. His instructions were to assist the envoy extraordinary and minister plenipotentiary, Walter Titley, and to keep the British government well informed of passing events. He seems to have performed his duties with regularity, tact, and ability, and on the death of Titley (27 Feb. 1768) he succeeded to the post of envoy extraordinary and minister plenipotentiary.
On 13 April 1771 he was appointed envoy extraordinary and minister plenipotentiary to the court of Prussia, but did not leave Copenhagen until the end of June, reaching Berlin in the following month. On 13 Dec. he was transferred with the same rank to the court of Russia, where he arrived early in the following June, and was received in the most distinguished manner by the empress. His instructions, dated 28 May 1772, directed him to offer the services of the British government as mediator between Russia and the Porte, with a view to effecting a treaty of peace, and to support the policy of the empress in Poland, but to attempt to secure toleration for the Greek church and other dissident religious bodies.