The family writes in a statement that Norman died after a short illness.
The composer turned 94 years old.
Norman's most famous work is the play that is played in the introduction to 25 films about the British agent James Bond.
Before the first Bond film "Agent 007 with the right to kill" in 1962, Monty Norman was contacted to write the play.
The filmmakers later hired composer John Barry to write a new arrangement for the piece.
Barry later claimed that he had in fact written the famous theme, but Norman won a defamation suit against The Sunday Times in connection with the newspaper making similar claims in an article.
According to the court, Norman was the author of the piece.