The Lowry theatre and arts centre in Manchester, where he was artistic director, said Mr Robson died suddenly on Saturday.
Mr Robson had been theatres' director at the Lowry since 1998. In 2003 he was appointed the centre's artistic director, taking strategic control of its artistic vision, encompassing performing and visual arts, and its community and education programme.
Born in Hamilton in 1954, and brought up in Motherwell, North Lanarkshire, Robson attended Hamilton Academy, before studying English and Drama at Glasgow University.
The Lowry chief executive Julia Fawcett said: "I was so impressed by his fierce intelligence, his passionate commitment to the arts and his independence of mind.
"Robert was respected and loved by all his colleagues.
"To me, he was indispensable: a calming influence at times of crisis, a giver of wise counsel, a trusted friend, a ferocious defender and advocate of the arts and an unparalleled source of industry gossip."
He is survived by his wife, Annette, and sons Stuart and Alan.
Cumbernauld Theatre artistic director Ed Robson said: "Robert, alongside David Taylor (now Portfolio Manager at Creative Scotland) together made Cumbernauld Theatre a powerhouse in the arts."