During Glenn's childhood he was regularly ill, and was, for a year, bed-ridden. Through intense training programs he got over his illnesses and a limp that he had had.

After graduating from a Pittsburgh high school, Glenn entered College of William and Mary where he majored in English. He then joined the Marines for three years and worked roughly five months as a reporter for the Kenosha Evening News. He then tried to become an author but found he could not write good dialogues and to get over it started going to acting classes.

In 1966, Glenn went to New York and joined George Morrison's acting class. He helped direct student plays to pay for his studies and appeared onstage in La MaMa Experimental Theatre Club productions, during this time. In 1967, he converted to Judaism and married Carol Schwartz, his wife up to now. In 1968, he joined The Actors Studio and began working in professional theatre and TV. In 1970, director James Bridges offered him his first movie role in The Baby Maker, released the same year.

Glenn that year left for LA and spent about 8 years there acting small roles in films and doing brief TV stints. He appeared in Francis Ford Coppola's Apocalypse Now (1979), in a small role, while there and also worked with directors like Jonathan Demme and Robert Altman. Fed up with Hollywood, in 1978 Glenn left LA with his family for Ketchum, Idaho and worked for the some two years he lived there as a barman, huntsman and mountain ranger, occasionally acting in Seattle stage productions.

In 1980, Glenn got back into acting in films, by appearing as ex-convict Wes Hightower in Bridges's Urban Cowboy. After, he appeared in action films like Silverado (1985), and The Challenge (1982) and drama films like The Right Stuff (1983), TV film Countdown to Looking Glass (1984), The River (1984) and Off Limits (1988) as he alternately played good guys and bad guys during the 1980s. He tried his hand at gangster movies in 1987 when he starred as the real-life sherriff turned gunman Verne Miller in the movie of the same name. "Verne Miller" was only given a theatrical release in Finland(!) and went straight to video in the U.S. In the beginning of the 1990s his career was at its peak as he appeared in several well-known films such as The Silence of the Lambs (1991), The Hunt for Red October (1990), and The Player (1992). Later he gravitated toward more different movie role, such as in black Freudian farce Reckless (1995/I), tragicomedy Edie and Pen (1997) and Ken Loach's socio-political declaration Carla's Song (1996). Today Glenn alternates mainstream films (Courage Under Fire (1996), Absolute Power (1997)), with independent projects (Lesser Prophets (1997) and Larga distancia (1998), written by his daughter Dakota Glenn) and TV (Naked City: A Killer Christmas (1998).