Fuller's parents were Jamaican and died when he was young; he was raised in an orphanage as a result. While in Detroit he was a schoolfriend of Paul Chambers and Donald Byrd, and also knew Tommy Flanagan, Thad Jones and Milt Jackson. After army service between 1953 and 1955 (when he played in a band with Chambers and brothers Cannonball and Nat Adderley), Fuller joined the quintet of Yusef Lateef, another Detroit musician. In 1957 the quintet moved to New York, and Fuller recorded his first sessions as a leader for Prestige Records.
Alfred Lion of Blue Note Records first heard him playing with Miles Davis in the late 1950s, and featured him as a sideman on record dates led by Sonny Clark and John Coltrane; Fuller's work on the latter's Blue Train album is probably his best known recorded performance. Fuller led four dates for Blue Note, though one of these, an album with Slide Hampton, was not issued for many years. Other sideman appearances over the next decade included work on albums under the leadership of Bud Powell, Jimmy Smith, Wayne Shorter, Lee Morgan and Joe Henderson (a former room mate at Wayne State University in 1956).
Fuller was also the first trombonist to be a member of the Art Farmer-Benny Golson Jazztet, later becoming the sixth man in Art Blakey's Jazz Messengers in 1961, staying with Blakey until 1965. In the early 1960s, he recorded two albums as leader for Impulse! Records, having also recorded for Savoy Records and Epic after his obligations to Blue Note had ended. In the late 1960s, he was part of Dizzy Gillespie's band. He went on to tour with Count Basie and also reunited with Blakey and Golson. Along with continuing to perform and record, Fuller is currently a faculty member of the New York State Summer School of the Arts (NYSSSA) School of Jazz Studies (SJS).
On January 13, 2010, Curtis' wife, Catherine Rose Driscoll Fuller, died.

John P. La Barbera is a Grammy® nominated composer/arranger whose writing spans many styles and genres. His works have been recorded and performed by Buddy Rich, Woody Herman, Count Basie, Dizzy Gillespie, Mel Torme, Chaka Khan, Harry James, Bill Watrous, and Phil Woods just to name a few. Though his major output has been in jazz, he has had works performed and recorded for symphony orchestra, string chamber orchestra, brass quintet, and other diverse ensembles. Most recently, Mr. La Barbera was chosen from among dozens of applicants to participate in the Jazz Composers Orchestra Institute at UCLA. As a result, John was one of sixteen composers commissioned by the JCOI to compose new works that meld jazz and symphonic music. "Morro da Babilonia" was the resulting work and was presented by the American Composers Orchestra in New York City at Columbia University's Miller Hall. His "Drover Trilogy" for string orchestra and corno da caccia was recorded by the late Dr. Michael Tunnell and has recently been released on Centaur Records. John's Grammy® nominated big band CD On the Wild Side along with Fantazm and his latest Caravan on the Jazz Compass® label, have been met with tremendous artistic and commercial success and are on the way to becoming jazz big band standards. As co-producer and arranger for The Glenn Miller Orchestra Christmas recordings (In The Christmas Mood I & II) John has received Gold & Platinum Records and his arrangement of "Jingle Bells" from those recordings can be heard in the Academy Award winning film La La Land. John is also the co-founder of the Diva Jazz Orchestra, the all women jazz big band founded in New York City in 1992. Mr. La Barbera is a Professor Emeritus of Music at the University of Louisville's School of Music and an international clinician/lecturer whose topics range from composing/arranging to intellectual property and copyright. Among his numerous organizational affiliations are Jazz Education Network, ISJAC, Phi Mu Alpha Sinfonia, NARAS, American Composers Forum, Chamber Music America, and a writer/publisher member of ASCAP since 1971.
John is a two-time recipient of The National Endowment for The Arts award for Jazz Composition and has served as a panelist for the NEA in the music category. His career has recently been profiled in "Bebop, Swing and Bella Musica: Jazz and the Italian American Experience" and in dozens of publications and encyclopedias. John's published works are considered standards in the field of jazz education.