Farmer was born an hour after his twin brother, on August 21, 1928, in Council Bluffs, Iowa, reportedly at 2201 Fourth Avenue. Their parents, James Arthur Farmer and Hazel Stewart Farmer, divorced when the boys were four, and their steelworker father was killed in a work accident not long after this.:443 Addison moved with his grandfather, grandmother, mother, brother and sister to Phoenix, Arizona when he was still four.:1–3
Farmer and his brother moved to Los Angeles in 1945, attending the music-oriented Jefferson High School, where they got music instruction and met other developing musicians such as Sonny Criss, Ernie Andrews, Big Jay McNeely, and Ed Thigpen. The brothers earned money by working in a cold-storage warehouse and by playing professionally.
By late 1945, Farmer was with Johnny Alston and His Orchestra recording for the Bihari Brothers' Modern Music label, backing Jeanne Demetz and, shortly after, on the Blue Moon label. Other band members for those recording dates included Al "Cake" Wichard and King Fleming. He later recorded with Teddy Edwards's band. Farmer played in several groups with his brother, including in ensembles led by Benny Golson and Gigi Gryce. He also played with Mose Allison, Jay McShann, Charlie Parker, and Miles Davis. He recorded extensively for Prestige Records.
Farmer died from sudden unexpected death syndrome on February 20, 1963, in New York City at the age of 34.
With Mose Allison
- Local Color (Prestige, 1957)
- Young Man Mose (Prestige, 1958)
- Ramblin' with Mose (Prestige, 1958)
- Creek Bank (Prestige, 1958)
- Autumn Song (Prestige, 1959)
- I Don't Worry About a Thing (Atlantic, 1962)
- Swingin' Machine (Atlantic, 1963)
With Gene Ammons
With Bob Brookmeyer
- Kansas City Revisited (United Artists, 1958)
With Teddy Charles
- Word from Bird (Atlantic, 1957)
- Coolin' (New Jazz, 1957)
- The Prestige Jazz Quartet (Prestige, 1957)
- Jazz In The Garden At The Museum Of Modern Art (Warwick, 1960)
With Art Farmer
- Early Art (New Jazz, 1954)
- When Farmer Met Gryce (Prestige, 1954–55)
- Art Farmer Quintet featuring Gigi Gryce (Prestige, 1955)
- Bennie Green with Art Farmer (Prestige, 1956)
- Farmer's Market (Prestige, 1956)
- Three Trumpets (Prestige, 1957) with Donald Byrd and Idrees Sulieman
- Last Night When We Were Young (ABC-Paramount, 1957)
- Portrait of Art Farmer (Contemporary, 1958)
- Modern Art (United Artists, 1958)
- The Aztec Suite (United Artists, 1959)
- Meet the Jazztet (Argo, 1960) - with Benny Golson
- Curtis Fuller and Hampton Hawes with French Horns (Status, 1957 )
With Stan Getz
- The Soft Swing (Verve, 1957)
With Teo Macero
- Teo (Prestige, 1957) - with the Prestige Jazz Quartet
With Sahib Shihab
- The Jazz We Heard Last Summer (Savoy, 1957)
With Mal Waldron
- Balliett, Whitney (September 23, 1985) "Profiles: Here and Abroad" The New Yorker, pp. 43–55.
- Ramsey, William E. & Shrier, Betty Dineen (2002) Silent Hills Speak: A History of Council Bluffs Barnhart Press. Cited in: Longden, Tom "Art Farmer" DesMoinesRegister.com
- Heckman, Don & Thurber, Jon (October 07, 1999) "Art Farmer: eloquent jazz master of the trumpet and fluegelhorn" Los Angeles Times.
- Balliett, Whitney (2006) American Musicians II: Seventy-One Portraits in Jazz. University Press of Mississippi.
- "Art Farmer: NEA Jazz Master (1999)" (June 29–30, 1995) Smithsonian Jazz Oral History Program NEA Jazz Master interview
- Bryant, Clora (1998) Central Avenue Sounds: Jazz in Los Angeles. University of California Press.
- Campbell, Robert L.; Pruter, Robert and Büttner, Armin "The King Fleming Discography" Archived May 21, 2009, at the Wayback Machine