He was educated at Westminster School and St Catharine's Hall, Cambridge, and soon showed a talent for drawing, and especially for humorous subjects. His more serious efforts did not rise to a high level, but his caricatures are as famous as those of his contemporaries Rowlandson. and Gillray, good examples being his "Country Club" (1788), "Barber's Shop" (1811) and "A Long Story" (1782.) He was a popular character, and the friend of most of the notabilities of his day, whom he never offended by attempting political satire; and his easy circumstances and social position (he was colonel of the West Suffolk Militia, and was appointed equerry to the duke of York in 1787) enabled him to exercise his talents in comfort.
His son, Sir Henry Edward Bunbury, bart (1778-1860), who succeeded to the family title on the death of his uncle, was a distinguished soldier, and rose to be a lieutenant-general; he was an active member of parliament, and the author of several historical works of value; and the latter's second son, Sir Edward Herbert Bunbury, also a member of parliament, was well known as a geographer and archaeologist, and author of a History of Ancient Geography.