Sir Robert Clermont Witt was the son of a wool merchant born in Hamburg whose mother was of Huguenot descent. He was educated at New College, Oxford, where he became an honorary Fellow in 1942. He served in the army during the Matabele (Ndebele) uprising in 1896 and acted as war correspondent with Cecil Rhodes. After returning to London he qualified as a solicitor and eventually became senior partner in Stephenson, Harwood and Tatham, making it one of the most prominent firms in London.
Witt was passionately interested in art. In 1902 he published How to Look at Pictures, which was reprinted several times. In 1903 he co-operated with Lord Balcarres (afterwards Earl of Crawford), Claude Phillips, D. S. MacColl and Roger Fry in founding the National Art Collections Fund. He was the first honorary secretary of the Fund (1903-20), then chairman (1920-45) and, on retirement, its first president. In his honour, a special exhibition of the principal acquisitions made for the nation by the Fund was organised at the National Gallery. Witt was also a trustee of the National Gallery (1916-23, 1924-31, 1933-40), trustee of the Tate Gallery (1916-31) and of the Watts Gallery, Compton, Surrey.
In 1932 he collaborated with Samuel Courtauld and Lord Lee of Fareham in founding the Courtauld Institute of Art, University of London and remained a member of its committee of management until his death. He bequeathed his collection of 6000 drawings to the Courtauld Institute, including a number by or attributed to Wilson.