Moss debuted for
Claremont Football Club in the WAFL in 1969. In 1970, he made his debut for the
Western Australian state team.
Essendon attempted to lure him to
Melbourne several times, and finally succeeded before
the 1973 season. In his first game, the opening round at
Windy Hill against Richmond, Moss made an immediate impression, appropriately enough following an austere
pre-game ceremony when the Essendon club held a minute's silence in honour of past legend
John Coleman who died some days earlier, and who also famously made a sensational debut. Moss played 89 games for Essendon, winning the
club's best and fairest three times, in 1974–76. He also represented
Victoria 5 times.
In 1976, he captained Essendon and won the Brownlow, but also suffered a serious knee injury. Moss has the second-highest average of Brownlow Medal votes (0.95 per game) of any player ever polled for the award. He was appointed captain-coach by Claremont
in 1977, and commented that his return from Essendon to Claremont was made easy by the fact the VFL and WANFL were of a similar standard at the time. That same year he won the
Simpson Medal, while playing for Western Australia against Victoria. Moss led Claremont to a premiership, their first in 17 years,
in 1981, and apart from the first two and last two years his tenure with the Tigers was their first period of consistent success since the days of
Johnny Leonard’s coaching. Moss ceased playing
in 1983 but returned for a few matches
in 1985, and retired as coach at the end of the 1986 season to be replaced by
Gerard Neesham. The
Graham Moss Medal has been awarded to the best on the ground in the WAFL State of Origin matches from 1995. In 1996, Moss was inducted into the
Australian Football Hall of Fame. In 2004, he was inducted into the
West Australian Football Hall of Fame and elevated to the status of "Legend" in 2006. ==Outside football==