He studied music composition at the University of Toronto, winning the Glenn Gould Composition Scholarship in 1985. Danna also served for five years as composer-in-residence at the McLaughlin Planetarium in Toronto (1987–1992). Works for dance include music for Dead Souls (Carbone Quatorze Dance Company, directed by Gilles Maheu 1996), and a score for the Royal Winnipeg Ballet's Gita Govinda (2001) based on the 1000-year-old classical Gita Govinda, with Choreographer Nina Menon. In June 2014, Danna was awarded an honorary Doctorate by the University of Toronto, for his career achievements in the field of music.

Danna was born in Winnipeg, Manitoba, but his family moved to Burlington, Ontario when he was four weeks old. He is the brother of fellow Composer Jeff Danna. He has been scoring films since his 1987 feature debut for Atom Egoyan's Family Viewing, a score which earned Danna the first of his thirteen Genie Award nominations. He has won five times for Achievement in Music - Original Score. Danna is recognized as one of the pioneers of combining non-Western sound sources with orchestral and electronic minimalism in the world of film music. This reputation has led him to work with such Directors as Atom Egoyan, Deepa Mehta, Terry Gilliam, Scott Hicks, Ang Lee, Gillies MacKinnon, James Mangold, Mira Nair, Billy Ray, Joel Schumacher, and Denzel Washington. His Soundtrack for Ang Lee's Life of Pi earned two Academy Award nominations for Best Original Score and Best Original Song Pi's Lullaby.