Michael Philip Mossman was born in Philadelphia on October 12, 1959 and went through the public schools of suburban Delaware County, graduating from Ridley High School. A National Merit Scholar, he took a double course at Oberlin College and the Oberlin Conservatory, finishing cum laude with degrees in sociology and anthropology alongside orchestral trumpet and jazz composition, and studied trumpet with Vincent Cichowicz of the Chicago Symphony Orchestra. He had been working professionally since he was seventeen. He later took a master's degree at Rutgers University, where he studied film scoring with Don Sebesky, a training in orchestration that shows up in almost everything he has done since.

The first decade of his working life was spent in the avant-garde rather than in the hard bop he would become known for. In 1978, at eighteen, he was in the trumpet section of Anthony Braxton's Creative Orchestra for a European tour, and the concert taped at the Grosser Sendesaal of WDR Cologne on May 12 of that year was eventually issued by hat ART in 1995. The following June he was in Paris for two Moers Music dates in a single month, Roscoe Mitchell's Sketches from Bamboo and Leo Smith's Budding of a Rose, and he toured with Mitchell into the early 1980s, returning to Milan in June 1983 for Roscoe Mitchell and the Sound and Space Ensembles, where he plays trumpet, piccolo trumpet, flugelhorn and percussion across five of the record's six pieces. He was also taking commercial session calls in the same years, among them the horns on Styx's Kilroy Was Here in 1983.

The mid-1980s turned him toward the mainstream. He played with Lionel Hampton in 1984 and spent a month in Art Blakey's Jazz Messengers, held the lead trumpet chair in Machito's orchestra, and worked with Gerry Mulligan in 1985. That same year he became musical director of Out of the Blue, the sextet Blue Note assembled from its young players, and made four albums with the band: O.T.B., cut at the Van Gelder Studio on June 7 and 8, 1985 with Kenny Garrett and Ralph Bowen in the reeds; Inside Track, recorded digitally at RCA Studio C on June 19 and 20, 1986; Live at Mt. Fuji, taped at the Mt. Fuji Blue Note Festival on August 31, 1986; and Spiral Staircase, made at A&R Studios on January 24 and 25, 1989 with Steve Wilson and Renee Rosnes in the band and four of its seven tunes written by Mossman. In 1986 he also appeared with the Naumburg Orchestral Concerts in the Central Park bandshell. He then joined Horace Silver's group for the run from 1989 to 1991, taped the sessions that became Rockin' with Rachmaninoff in June and August 1991, played lead trumpet in Toshiko Akiyoshi's Jazz Orchestra, and joined Mulligan's Rebirth of the Cool band in 1992. He toured with Dizzy Gillespie's United Nation Orchestra, with Slide Hampton's Jazz Masters Orchestra, and with the Philip Morris Superband, whose Town Hall concert of September 23, 1989 was released under Gene Harris's name.

Running underneath all of that is the Latin jazz career that has defined him longest. After Machito came Mario Bauzá's Afro-Cuban Jazz Orchestra, for which he both played and wrote: "Paz Profundo" on My Time Is Now (1993) is his composition and his arrangement, and on 944 Columbus (recorded at Clinton Studios on May 27 and 28, 1993) he takes the trumpet solo on four of the ten pieces. He worked with Tito Puente, Eddie Palmieri and Chico O'Farrill, and he was a fixture of Ray Barretto's New World Spirit, playing trumpet, trombone and flugelhorn on My Summertime (1995) and arranging two of its tracks, then writing "Moss Code" and "Dance of Denial" for Contact! (1997), which was nominated for Best Latin Jazz Album at the 41st Grammy Awards. With Michel Camilo he goes back to On the Other Hand (1990) and One More Once (1994) and forward to the concert taped in the Dominican Republic on December 3, 1994 that Calle 54 issued as Caribe. Since the mid-2000s he has been lead trumpet in Arturo O'Farrill's Afro-Latin Jazz Orchestra at Jazz at Lincoln Center, on Una Noche Inolvidable (recorded at Rose Hall on January 28 and 29, 2005, with two of his arrangements) and on Song for Chico, taped on April 23, 2006, which won the 2009 Grammy for Best Latin Jazz Album.

His own catalogue is small and unusually varied. It opens with two co-led quintet records: Uptown Dance (EGT, 1992), shared with the Valencian alto saxophonist Perico Sambeat and filled out by David Kikoski, Bill Moring and Keith Copeland, and Granulat (Red Record, 1991), made with the Swiss reed player and composer Daniel Schnyder at the Barigozzi Studio in Milan in April 1990, on which Mossman plays trumpet, piccolo trumpet and flugelhorn. Springdance (Claves Jazz), recorded on May 9 and 10, 1994, is the first record fully under his own name: eight Mossman originals for a quintet of David Sanchez, Kenny Drew Jr., James Genus and Marvin "Smitty" Smith with Bobby Sanabria on percussion, and a coupling of Bill Evans's "Peace Piece" with Horace Silver's "Peace." Mama Soho, taped at Systems Two on December 3 and 4, 1995 and issued by TCB in 1998, put Bob Mintzer's tenor beside him in a sextet with the same rhythm section. The Orisha Suite (Connector Music, 2001) is his Afro-Cuban song cycle, written and arranged throughout, with Arturo O'Farrill on piano and organ, John Benitez, Horacio "El Negro" Hernandez, Adam Rogers and Todd Williams in the band and Paquito D'Rivera, Ray Barretto and Carlos "Patato" Valdes appearing as guests. Misa Afro Cubana (2007) sets a mass for the WDR Big Band of Cologne with the Cuban singer Xiomara Laugart and the trombonist Juan Pablo Torres. In 2010 he premiered Visitatio Sepulchri de Gandia with a Valencian septet, and in 2021 he wrote, conducted and played on Colors & Shadows, a WDR Big Band programme built around the Catalan trumpeter and singer Andrea Motis.

The arranging is not a sideline to the trumpet playing but the other half of the career, and it has been the source of most of his awards. He arranged and conducted "Recordame" on Joe Henderson's Big Band, which won Best Large Jazz Ensemble Performance at the 40th Grammy Awards. He wrote three of the ten arrangements on the Mingus Big Band's ¡Que Viva Mingus! (1997), conducting his own charts of "Tijuana Gift Shop," "Los Mariachis" and "Far Wells Mill Valley," and returned for "Freedom" and "Meditations for a Pair of Wire-Cutters" on Blues & Politics (1999) and an adaptation of "Invisible Lady" on Tonight at Noon (2002). His "Afro-Cuban Jazz Suite for Ellington," recorded by the Bobby Sanabria Big Band on Multiverse, brought him a nomination for Best Arrangement, Instrumental or A Cappella at the 55th Grammy Awards in 2013; he had already written and arranged "57th St. Mambo" for Sanabria's Big Band Urban Folktales and "Mosscode" for Afro-Cuban Dream: Live and in Clave. In 2019 he wrote every arrangement on Michel Camilo's eighteen-piece big band record Essence, on which he is also a trumpet soloist. He is a regular guest arranger and conductor with the German radio orchestras, responsible for Mambo Nights with Arturo Sandoval and the WDR Big Band, Latin Jazz Latino with Joe Gallardo and the NDR Bigband (2005), Spirits Dancing with David Sanborn and the hr-Bigband (2009), and the WDR charts behind Hendrik Meurkens's Samba Jazz Odyssey (2022). He has also conducted the Bilbao Orkesta Sinfonica in his own works, written the ballets Beneath the Mask, danced at the Harris Theater in Chicago in 2006 by the Deeply Rooted Dance Company with Jon Faddis and the Chicago Jazz Ensemble, and La Cova do Rey Cintolo, premiered in Spain in 2010. His arrangement of Faddis's "Teranga" was played by the Philadelphia Orchestra at the Kimmel Center in 2006 and his arrangement of Paquito D'Rivera's "I Remember Diz" by the Louisiana Philharmonic in 2010, and his "Latin Tinge" was performed at the Kennedy Center in 2010 with D'Rivera and the Quartet Indigo string quartet. For Fernando Trueba he has scored Two Much (1995), Calle 54 (2000), in which he also appears on screen, and the Academy Award nominated animated feature Chico and Rita (2011).

He has stayed a working section player through all of it. He was one of five trumpets in the Jimmy Heath Big Band on Turn Up the Heath (2006) and again on Togetherness, recorded live at the Blue Note on October 30 and 31, 2011 beside Roy Hargrove, Diego Urcola, Frank Greene and Greg Gisbert. He is in the trumpet section of Joris Teepe's big band on We Take No Prisoners (2009), plays on Antonio Hart's All We Need (2004) and on Arturo O'Farrill's record with Claudia Acuña, In These Shoes (2008), and spent much of the 1990s in Bob Mintzer's big band. Alongside the playing he has built a long teaching career, with earlier faculty posts at the Oberlin Conservatory, Rutgers, The New School, Bloomfield College and Manhattanville College, and he is now Professor and Director of Jazz Studies at the Aaron Copland School of Music at Queens College, City University of New York, as well as a member of the jazz faculty at the Juilliard School.

Mossman describes himself in one sentence that explains most of his output: "I am at heart a hard bop trumpet player. I am also an arranger/composer and I tend to play like one." Scott Yanow's AllMusic entry calls him "a fine hard bop trumpeter with a wide range" and a flexible one, and flexibility is the trait his career has actually required, from Anthony Braxton's Creative Orchestra to the Machito band to the Blue Note front line to a mass written for a German radio orchestra. He names Freddie Hubbard's sound as foundational and Maynard Ferguson's "sound and fire" as something that runs deep in him, and outside the trumpet he points to Duke Ellington, Charles Mingus and Horace Silver, which is to say to three composers who wrote for specific players rather than for instruments. That is how he writes as well: his Mingus Big Band charts and his Essence arrangements for Michel Camilo rescore existing pieces around the soloists in the room rather than transcribing them upward for more horns. He resists a fixed approach, saying that "dealing with the music in the moment keeps things fresh and not formulaic and predictable," and he adapts his articulation and phrasing to the idiom in front of him instead of importing one manner into all of them, which is why the same player can hold a lead chair in an Afro-Cuban orchestra and take the solo on a Roscoe Mitchell free piece. The doubling matters too. He plays trombone on his own records and on several of Ray Barretto's, and the habit of hearing the middle of the horn section from inside it feeds directly into the density of his big band writing.