He was the son of Revd Dr John Burns, a minister of the Barony Church, and Elizabeth Stevenson.
In 1804 he went to London to seek medical service in the army, and was induced to go to St. Petersburg to take charge of a hospital about to be established by the Empress Catherine on the English plan; but finding the position uncongenial, he returned to Scotland in a few months.
He had failed to make arrangements for a salary, and on discovering that government surgeons were paid £90, he returned to Scotland, where he became a highly popular lecturer on anatomy[2]—wearing the diamond and topaz ring given to him by the Empress Catherine when he left Russia.
Allan authored a number of publications which were quickly translated into German and were published concurrently in the United States.
His favourite pupil, Granville Sharp Pattison, has a short memoir of him, prefixed to an addition of some of his writings, which were translated for text-books on the Continent.