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Corri was born in Rome and studied voice with Nicola Porpora in Naples. In 1781 he moved to Scotland with his family to become an opera conductor. He became an important singing master in Edinburgh. Both his wife and daughter (Sophia Giustina Corri), who was a talented singer and pianist, were pupils of his. He was manager of the Theatre Royal in Edinburgh, and also conducted the concerts of the musical society there.
Around 1790, Corri moved to London and began publishing vocal music in Soho. Corri had financial problems on several occasions. With a musician named James Sutherland and his son John Corri he founded a publishing business. When Sutherland died in 1790 the company ceased to exist. His son-in-law Jan Ladislav Dussek joined the company in 1794. When the company of Dussek and Corri went bankrupt, Dussek left England for Germany, leaving behind his family, and his father-in-law (Corri) apparently in a debtor's jail. The business was later taken over in 1804 by his son Philippe Corri. An apprentice to Corri in London was Isaac Nathan.
Domenico Corri was also the manager of the Vauxhall Pleasure Gardens in London. In 1812, Corri organised a concert in the Argyll Rooms in Regent Street, London. In this, the favorite concert hall in London at this time, Corri put together quite a contrasting program as the evening's entertainment: Giovanni Battista Pergolesi's Stabat Mater combined with a Ball. Corri died in London, aged 78.
Three of his children were musicians: Sophia Giustina Corri (1775–1831), who later used the name Sophia Corri Dussek, Philippe Corri (1784–1849), who later immigrated to Baltimore, MD using the name Arthur Clifton, and Haydn Corri (1785–1860). His brother Natale Corri, was also a composer. A niece, Fanny Corri-Paltoni, was a successful soprano. One of his descendants was puppeteer Christine Glanville.
- A Select Collection of the Most Admired Songs, Duetts, &c, Volumes I-III 1779[?], Volume IV 1795. Reprinted in C. R. F. Maunder, Domenico Corri's Treatises on Singing: A Select Collection of the Most Admired Songs, Duetts, etc. and the Singer's Preceptor : A Four-Volume Anthology. New York; London: Garland Publishing, 1995. [Volumes also include musical sources for the Select Collection & The Singer's Preceptor].
- A Complete Musical Grammer, 1787.
- A Musical Dictionary, 1798.
- The Art of Fingering, 1799.
- The Singer's Preceptor, or Corri's Treatise on Vocal Music. London: Chappell, 1810 or 1811. Reprinted New York/London, 1995. Reprinted in: The Porpora Tradition: Master Works of Singing. Edited by Edward Foreman, Vol. 3. Champaign, Illinois: Pro Musica Press, 1968.
- La Raminga Fedele, 1770
- Alessandro nell'Indie, 1774
- The Travellers, or Music's Fascination, Drury Lane, January 22, 1806
- Baker's Biographical Dictionary of Musicians, (Nicolas Slonimsky, Hrsg.) New York: G. Schirmer, 1958
- Oxford Companion to Music. Edited by Percy Scholes. London: Oxford University Press, 1963.