Abbott was born on 13 May 1763 at Cowick near Exeter, Devon. By profession a surgeon and apothecary, he was a keen amateur painter in both watercolour and oils. He studied in Exeter with Francis Towne, to whom he was also a friend and patron, and his watercolour style was based on Towne’s. His watercolours were generally landscapes, for which he toured to the Lake District, but in oil he also did history paintings.
Abbott exhibited regularly at the Royal Academy between 1795 and 1805; he is last recorded exhibiting there in 1822. His oil paintings were once well known but the great bulk of his work was landscape watercolours. He had a series of etchings of his paintings created which was nearly complete at the time of his death. He made a sketching tour to Scotland and the Lake District in 1791.
In 1825 he inherited the estate of Fordlands, near Exeter, and in 1835 he was appointed a Deputy Lieutenant for the County. It seems that Abbott never sold any of his paintings, and most of his works were retained by his family until well into the 20th century. Despite this, in his lifetime he was better known than his teacher, Francis Towne, some of whose Italian views he copied. Examples of his work are in the British Museum, Victoria & Albert Museum, Exeter Museum, NG Scotland & elsewhere

