In 1714 Parliament offer a £20,000 prize for anyone who can provide an accurate means of measuring longitude at sea. John Harrison (Michael Gambon) flies in the face of popular opinion by saying that the stars do not provide the answer, and provides his own solution with the invention of a mechanical clock. However, it takes Harrison forty years to prove his theory, and he is eventually forgotten in the mists of time. Centuries later, Robert Gould (Jeremy Irons) attempts to restore Harrison's reputation by tracking down and repairing the four clocks he originally constructed.

Luzhin (John Turturro), a Russian master visiting Italy for the World Chess Final, finds his world turned upside down when he unexpectantly falls in love. Despite his genius Luzhin is extremely introverted and finds it difficult to function in the everyday world, but when he meets Natalia (Emily Watson) she offers him a glimpse of a life outside the chess world and opens up the possibility of a whole new way of living. However, she is also about to get married.