Stevenson was born in Largs, Ayrshire, Scotland and studied in Dublin and Glasgow. He was a lecturer (1970-1980), senior lecturer (1980-1984), and reader Scotland, (1984-1990) at the University of Aberdeen. He was Professor of Scottish History at the University St. Andrews, Scotland (1991-1994), and has been honorary professor Scotland, since 1994. He was Chairman of the Company of Scottish History, 1991-1994.
He was a Member of the Scottish Records Advisory Council, Scotland, 1989-1994.