He attended the University of California, Berkeley, where he was a four-year letterman and two-year starter. He started his career at defensive tackle before moving to offensive tackle in his junior year. He was selected with the Colts' first pick (#19 overall) in the 1997 NFL Draft. He has started all 16 games in seven of his ten pro seasons, and only missed six games in 2003 with a knee injury. He made his first Pro Bowl appearance in 2005. The NFL told Glenn that he would also appear in the 2006 Pro Bowl, however they later informed him that he did not receive enough votes and that he would be an alternate. Glenn did end up participating in the 2006 Pro Bowl replacing the injured Willie Roaf. Tony Dungy, who generally backs the NFL, verbally stated that he was disappointed with the way in which the NFL handled this particular situation.
On July 24, 2007, Glenn announced his retirement. He said he lost his passion for football.
In August, 2011, Glenn became President of D.R.E.A.M. Alive, Inc., an Indianapolis-based non-profit organization founded by Glenn and his wife, Maya.