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 De Metz 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 the jazz label Prestige.
Farmer died suddenly 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)
With Teddy Charles
- The Prestige Jazz Quartet (Prestige, 1957)
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)
- 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 Teo Macero
- Teo (Prestige, 1957) - with the Prestige Jazz Quartet
With Mal Waldron
- Teddy Edwards (Cool 'n Blue, 1946-48)
- Gene Ammons: The Happy Blues (Prestige, 1956); All Star Sessions (Prestige, 1950-55)
- Sonny Criss: California Boppin´ (Fresh Sound, 1957)
- Teddy Charles: Coolin´ (Prestige, 1957)
- 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"