Cragg was born in Liverpool, and following a period of work as a laboratory technician, studied art in Cheltenham and the Royal College of Art. He left in 1977 and moved to Wuppertal in Germany, where he has lived and worked since.
Many of Cragg's early works are made from found materials, often industrial waste and plastic. One of his better known works, Britain as Seen From the North (1981), features the shape of the island of Great Britain on the wall, oriented so that north is to the left. To the left of the island is the figure of a man, apparently Cragg himself, looking at the country from the position of an outsider. The whole piece is made from broken pieces of found rubbish and is often interpreted as commenting on the economic difficulties Britain was going through at that time which had a particular effect on the north.
Later, Cragg used more traditional materials, such as wood, bronze and marble, often making simple forms from them, such as test tubes.
Cragg won the Turner Prize in 1988.