Anstey was the son of a wealthy clergyman the rector of Brinkley, Dr. Anstey in Cambridgeshire, where he was born. He was educated at Eton College and King's College, Cambridge, where he distinguished himself for his Latin verses. He became a fellow of his college (1745); but the degree of M.A. was withheld from him, owing to the offence caused by a speech made by him beginning: "Doctores sine doctrina, magistri artium sine artibus, et baccalaurei baculo potius quam lauro digni." In 1754 he succeeded to the family estates and left Cambridge; and two years later he married the daughter of Felix Calvert of Albury Hall, Herts. For some time Anstey published nothing of any note, though he cultivated letters as well as his estates. Some visits to Bath, however, where later, in 1770, he made his permanent home, resulted in 1766 in his famous rhymed letters, The New Bath Guide or Memoirs of the Blunderhead Family..., a satirical poem of considerable sparkle, about the adventures of the "Blunderhead" family in Bath, from which Tobias Smollett is said to have drawn largely in his The Expedition of Humphry Clinker. The work had immediate success, and was enthusiastically praised for its original kind of humour by Walpole and Gray. The Election Ball, in Poetical Letters from Mr Inkle at Bath to his Wife at Gloucester (1776) sustained the reputation won by the Guide. He made many other excursions into literature which are hardly remembered, and ended his days as a country squire at the age of eighty. His Poetical Works were collected in 1808 (2 vols.) by the author's son John (d. 1819), himself author of The Pleader's Guide (1796), in the same vein with the New Bath Guide.