Norman Adams studied at Harrow School of Art from 1940 to 1946 and subsequently at the Royal College of Art from 1948 to 1951. He was Head of Painting at Manchester College of Art from 1962 to 1970, visiting tutor at Leeds University from 1973 to 1976 and Professor of Painting at the University of Newcastle-upon-Tyne from 1981 to 1986. In 1986 he was elected Keeper of the Royal Academy in 1986, and upon retiring from this position after nine years, was appointed the Academy's Professor of Painting Emeritus in 1995. His first solo exhibition was held in 1952 at Gimpel Fils, London with subsequent regular shows held at Roland, Browse and Delbanco.

In 1955 Norman and his wife Anna bought a house in Horton in Ribblesdale where they spent much of the next fifty years together. In 1962 they paid their first regular visit to the west coast of Scotland, fascinated by the islands and the ever changing light. In 1953 he designed the stage set and costumes for 'A Mirror of Witches' produced by the Royal Ballet at Covent Garden, and again for the Sadler's Wells production of Saudades in 1955. Commissions have played a large part in Adams's career, his first being the painting of a mural for Broad Lane Comprehensive School, Coventry in 1954. In 1967 he was commissioned by the Oxford University Press to illustrate parts of the Old Testament. He went on to paint murals for St Anselm's Church, Kennington, London in 1971 and to make 14 ceramic panels of the Stations of the Cross for the Roman Catholic Church at Coffee Hall in Milton Keynes in 1975. In 1994 Adams received a commission for the Fourteen Stations of the Cross, for St Mary's Church, Mulberry Street, Manchester. These were exhibited at the Sackler Galleries, Royal Academy, prior to installation. His work can be found in numerous public collections including that of the Tate Gallery in London.

A retrospective of the artist's work was held in the Diploma Galleries, Royal Academy in 1988.