[3] Greenwood was born in Fife and educated at Dunfermline High School and Edinburgh University, where he read English.
[6] He became Scottish captain, but after losing 15–0 to France, lost that position to Angus Cameron, while keeping his place on the team.
Greenwood played 16 matches on the tour, and scored tries in the first and fourth tests of the series, which the Lions drew 2-2 with South Africa.
In a newspaper report at the time, he was described as "without equal among his contemporaries for a combination of skill, perception and deadly physical pace.
Greenwood was in contention for a place on the next Lions tour (in 1959), but suffered an injury (variously described as a broken collarbone or dislocated shoulder) playing against Ireland that season.
At Loughborough he influenced generations of rugby players, coaches and students including Clive Woodward, Andy Robinson, Fran Cotton, and Liza Burgess.
In 1998 Greenwood was one of the inaugural inductees into the National Coaching Foundation's Hall of Fame, and won the prestigious Geoffrey Dyson Award for his outstanding contribution.