Arthur Stewart Farmer was born on August 21, 1928, in Council Bluffs, Iowa, an hour ahead of his identical twin brother, the bassist Addison Farmer. Their parents divorced when the boys were four, and their father, a steelworker, was killed in a workplace accident soon afterward. The family moved to Phoenix, Arizona, where Farmer's grandfather served as a minister in the African Methodist Episcopal Church and his mother played piano for the choir. He began on piano, passed through bass tuba and violin, and settled on cornet and then trumpet at thirteen. Phoenix schools were segregated and no teacher at his school could help him, so he taught himself to read music. In 1945 the twins moved to Los Angeles and enrolled at Jefferson High School, a music-oriented school whose other students included Sonny Criss, Ernie Andrews and Ed Thigpen. Farmer was working professionally on trumpet by sixteen, in bands led by Horace Henderson, Jimmy Mundy and Floyd Ray, partly because so many older players were still in the armed forces.

An early tour with Johnny Otis ended after four months when the seven-nights-a-week schedule lacerated his lip and he could no longer play. He went to New York for technique training, worked as a janitor and freelanced through 1947 and 1948, failed an audition for Dizzy Gillespie's big band, and returned to the West Coast in 1948 with Jay McShann. Studio and club work in Los Angeles was largely closed to Black musicians in that period, but Farmer played and toured with Benny Carter, Roy Porter and Gerald Wilson before joining Wardell Gray in 1951. His first known studio date was in Los Angeles in mid-1948, backing the singer Big Joe Turner and pianist Pete Johnson for Swing Time Records. On January 21, 1952, a Wardell Gray sextet session produced "Farmer's Market," a Farmer original that gave him his first real visibility.

Farmer joined Lionel Hampton's orchestra in 1952 and toured Europe with it from September to December 1953, sharing the trumpet chairs with Clifford Brown, Quincy Jones and Benny Bailey. In 1953 he also joined Teddy Charles's New Directions band, whose written material pushed him toward a wider improvisational vocabulary. He relocated to New York and cut his first session as a leader on July 2, 1953; combined with a second date eleven months later, it became the Prestige LP The Art Farmer Septet, with arrangements by Quincy Jones and Gigi Gryce. He worked steadily with Gryce from 1954 to 1956, with Horace Silver's quintet from 1956 to 1958, and with Gerry Mulligan's pianoless quartet from 1958 to 1959. Label politics kept the Silver association to two records: Silver was a Blue Note artist and Farmer was signed to Prestige, and the two companies' owners did not lend players to each other. Farmer played in a Thelonious Monk sextet on a Steve Allen Show broadcast on June 10, 1955, and in the Charles Mingus sextet at the Newport Jazz Festival the following month. In New York, Lester Young told him to tighten up and tell a story in every solo, and for a period Farmer rented his horn by the night to Miles Davis, who had pawned his own. His reputation for being able to read and play anything put him on records by the leading arrangers of the day, George Russell and Quincy Jones among them, and in 1957 on a set of experimental sessions with the composer Edgard Varèse, who wanted the players to improvise inside approximate notation. Farmer was one of the fifty-seven musicians in Art Kane's 1958 photograph "A Great Day in Harlem," and was interviewed for the 1994 documentary built around it.

In 1959 Farmer and the tenor saxophonist and composer Benny Golson formed the Jazztet, each having independently decided the other should be in his new sextet. The band ran until 1962, recorded for Argo and Mercury, and gave early exposure to the pianist McCoy Tyner and the trombonist Grachan Moncur III. Farmer then built a group around the guitarist Jim Hall and the bassist Steve Swallow, a partnership that ran from 1962 to 1964 and included two European tours, one of which the BBC filmed for its Jazz 625 series. Hall left partway through the second tour while the group was working in Berlin; a pianist took his place, and after some churn that chair went to Steve Kuhn, the lineup that recorded Sing Me Softly of the Blues for Atlantic in 1964. Farmer toured Europe again in 1965 and 1966, came home and co-led a small group with Jimmy Heath, and, as rock cut into the work available to jazz players, spent six months in Elliot Lawrence's pit orchestra for the Broadway production of The Apple Tree.

Farmer moved to Europe in 1968 and settled in Vienna. He played with the Kenny Clarke / Francy Boland Big Band and joined the Austrian Radio Orchestra, a post that at first asked only ten days a month and left him free to work with other expatriates in Europe including Don Byas, Dexter Gordon and Ben Webster. As the orchestra's repertoire drifted away from jazz and took more of his time, he left after three or four years. He toured constantly from his Vienna base, telling an interviewer in 1976 that he was traveling ninety percent of the time and could live anywhere so long as he could reach an airport. A 1982 revival of the Jazztet with Golson brought him back to American stages far more often than he had been in the previous decade, and through the 1980s he also led an international touring quintet featuring the tenor saxophonist Clifford Jordan. He gave up smoking and drinking in the early 1980s and lost thirty pounds, changes he described to The New Yorker in 1985 as the difference between needing a drink to play and being unable to do both.

From the early 1990s Farmer kept a second house in New York and split his time between the two cities, with regular runs at the Sweet Basil Jazz Club alongside Clifford Jordan and later at the Village Vanguard with Ran Blake and Jerome Richardson. Austria awarded him its Gold Medal of Merit in 1994, the year a concert in his honor was held at Alice Tully Hall in New York. He recorded prolifically to the end, including orchestral projects with American and European ensembles, and reviewing Silk Road from 1996 the critic Scott Yanow judged the sixty-eight-year-old Farmer to be still in his prime. In 1999 the National Endowment for the Arts named him a Jazz Master. A few months after that, on October 4, 1999, Farmer suffered a fatal heart attack at his home in Manhattan. He was seventy-one, and was survived by his companion and manager Lynne Mueller and by his son Georg.

Farmer's first models on trumpet were Dizzy Gillespie, Miles Davis and Fats Navarro, but hearing Freddie Webster redirected him: he decided to work on sound while, as he put it, everyone his age was working on speed. That decision defines the playing. He avoided the bright, penetrating attack of orthodox trumpet in favor of a rounded, unhurried lyricism, letting the beat run slightly ahead of him in a way The Guardian compared to Billie Holiday's phrasing, and the switch to flugelhorn in the early 1960s let him get the warmth he wanted without reaching for a mute. Alongside Clark Terry he did more than anyone to establish the flugelhorn as a solo voice in jazz rather than a section color, and in 1989 he worked with the instrument maker David Monette to create the flumpet, a trumpet and flugelhorn hybrid he played almost exclusively in his last decade. Critics recognized the individuality early: writing about a 1957 Hal McKusick date, Whitney Balliett called Farmer one of the few genuinely individual modern trumpeters at a time when most were copying Gillespie or Davis. He never settled, moving from bebop through the composed frameworks of George Russell and Teddy Charles, absorbing the harmonic language of John Coltrane's Giant Steps period, and playing melodious, unhurried music through the years when the avant-garde dominated the conversation. A reviewer hearing him at sixty-seven still described a style that was continuing to evolve.