Robbins began his acting career in 1984, appearing in two episodes of Prisoner, and in 1985 Robbins appeared on the sketch show The Eleventh Hour. He shot to fame in 1988 when he starred in another sketch show, The Comedy Company, in which one of his characters, 'Uncle Arthur', became very popular. In 1991, he joined the team of Fast Forward for two seasons. Other sketch shows in which he has appeared include Full Frontal (1993), Jimeoin (1994–95) and Something Stupid (1998).
In 1998, Robbins became a panellist on the weekly television show The Panel. This aired on Channel Ten. The show lasted eight seasons, before ending in 2005.
Robbins played Russell Coight in All Aussie Adventures from 2001 until 2002 on Channel Ten. In 2001 Robbins played the role of Pete O'May in the award-winning Australian film Lantana. From 2002 to 2012, he played the role of Kel Knight in the television comedy series Kath & Kim. He also appeared in the Da Kath and Kim Code telemovie in 2005 and the film Kath and Kimderella in 2012.
In 2006, Robbins appeared in two episodes of Channel 10s Thank God You're Here. He also co-starred with Mick Molloy in the comedy feature BoyTown as an eighties' pop-star called Benny G. making an ill-advised comeback; and was also executive producer of the Network 10 sketch comedy show The Ronnie Johns Half Hour. In 2008, Robbins launched a new game show on Channel 7, entitled Out of the Question. In 2011, he was the guest on the fourth episode of the comedy program, The Joy of Sets with Tony Martin and Ed Kavalee on Channel 9. Also in 2011, he appeared in the comedy Scumbus - coincidentally, written by Kavalee. In 2013, he appeared in ABC1's sitcom, Upper Middle Bogan, which has been renewed for a second season of eight episodes.
Glenn has appeared once on The Little Dum Dum Club, on the show's 29th episode back in 2011.