Sir Rex maintained an interest in the art world as well; he was a member of the National Gallery Scientific Advisory Committee from 1978 to 2007 and its Chairman from 1991–1993. In 1981, Richards became a founding member of the World Cultural Council. He was Trustee of the Tate Gallery from 1982–1988 and 1989–1993, of the National Gallery from 1982–1988 and 1989–1993, and of the Henry Moore Foundation from 1989–2002; he was Chairman of the Moore from 1994–2001. He was also Chairman of the British Postgraduate Medical Foundation from 1986–1993.