Joseph Norman Alessi was born in Detroit, Michigan and attended high school in San Rafael, California.

His father, also named Joseph Alessi, was a professional trumpet player, and his mother, Maria (née Leone) sang in the Metropolitan Opera chorus.

Alessi made his solo debut with the New York Philharmonic in 1990 when he performed the Fantasy for Trombone by Paul Creston.

In 1992, he performed with the New York Philharmonic in the world premiere of the Trombone Concerto by Christopher Rouse, which later received the 1993 Pulitzer Prize for Music.

Joseph Alessi's playing is often noted for refined musicianship, an extraordinarily rich sound quality and complete technical control.