Cosmo Alexander was the son and pupil of John Alexander, an Aberdonian painter and engraver. Cosmo was a portrait painter; however, the defining factor in his life was his belief in the Jacobite cause. He participated in the 1745 Rising, after which he went to Rome to seek refuge. He stayed there from 1747 to 1751, painting portraits of members of the expatriate Jacobite community, including the family of Bonnie Prince Charlie. He travelled throughout Italy and visited Paris before settling in London in 1754. A decade later, he visited the Netherlands, and in 1766, he embarked on an extended tour of the American colonies, where he once again found favour painting portraits of émigré Scots. He was the first teacher of the artist Gilbert Stuart, who he returned to Scotland with in 1771.