Sir James Burrow FRS FSA (28 November 1701 – 5 November 1782 at Starborough Castle, Lingfield, Surrey) was an English scholar and legal reporter at Inner Temple, London, and was vice president and twice briefly president of the Royal Society.

He was elected a Fellow of the Royal Society on 7 April 1737, as "A Gentleman well versed in Natural and Mathematical knowledge".

He twice served briefly as a president of the Royal Society, from October to November 1768 following the death of The Earl of Morton, and July to November 1772, following the death of James West.

As vice president, he was involved in the society's activities in organising the observation of the 1761 Transit of Venus, signing the Articles of Agreement between the Council of the Royal Society and Mr Charles Mason and Mr Jeremiah Dixon for their expedition to Bencoolen in the Island of Sumatra.

At the time, four reporters were formally appointed by the King 'to commit to writing, and truly to deliver, as well the words spoken, as the judgments and reasons thereupon given,' in the courts of Westminster quoted in [1].