London in the 1950s. Dame Lettie Colston receives a mysterious phone-call in which the caller says simply, 'Remember you must die'. Disturbed by this, she goes to stay for a few days with her brother, Godfrey and his wife Charmian, a once successful novelist whose work has recently come back into vogue. Charmian, however, is getting a little forgetful and is missing her former maid, Jean Taylor, who is now in hospital.

Having intercepted another threatening phone-call to Lettie, Godfrey is shocked to read about the death of Lisa Brooke, with whom he once had a secret affair. At Lisa's cremation service, Godfrey and Lettie are joined by Lisa's former companion, Mrs Pettigrew, and also by the literary critic Guy Leet and the poet Percy Mannering, who has been conducting a long-running literary feud with Leet over the latter's dismissal of him as no more than a competent versifier. Later, as is their regular custom, Mannering's granddaughter Olive entertains Godfrey by exposing her stocking-tops to him in return for payment. She uses this money to help her grandfather and also occasionally to assist Godfrey's son, Eric, a peevish and struggling writer living in the shadow of his more talented mother and spying on his father's activities.

The mystery phone-calls continue and now begin to include Godfrey, Charmian, Leet, Mannering and Lottie's circle of acquaintances. A friend of Charmian's, ex-Inspector Mortimer, is called in to investigate and he questions a number of people, including Jean Taylor, who, introducing him to her hospital friends, also supplies him with information about the recipients of the phone-calls. Meanwhile, Mrs Pettigrew, left only a paltry sum in Lisa Brooke's will, is on the lookout for a new position, and Lottie recommends her as housekeeper to Charmian, who not only dislikes her but also feels vaguely threatened. However, her attempts to persuade Godfrey to dismiss her are unsuccessful, because, unknown to Charmian, Mrs Pettigrew is blackmailing him with letters that reveal his previous affair with Lisa Brooke.

One lunchtime, Mortimer calls together all the old people who have had similar threatening phone-calls, but the meeting descends into chaos before he can disclose his solution. Heartened by Mortimer's obvious affection for her and admiration for her work, and tired of Godfrey's indecision over Mrs Pettigrew, Charmian decides to move into a nursing home. Fearing what might happen if Godfrey is left in the clutches of Mrs Pettigrew, Jean Taylor asks Mortimer to pass on a message from her to Godfrey, which reveals that Charmian had once had an affair with Leet. When Godfrey visits Charmian at the home and smugly reveals that he knows about her former liaison with Leet, he is startled, then relieved, to find that she has known about his affair with Lisa Brooke for years.

While his father is away, Eric Colston has met Mrs Pettigrew and threatened to tell his mother about Godfrey's infidelity if she does not cut him in on her blackmailing scheme. She reluctantly agrees and they await Godfrey's return. They are taken aback by his cheerfulness, however. No longer fearing exposure, Godfrey dismisses their attempts to extort money from him and orders them both out of the house. At that point he is visited by the police, to be told that Lottie has disturbed a masked burglar in her home and has been murdered.

The murderer is discovered to be the boyfriend of Lettie's former maid. He has panicked when Lettie appeared to recognise him. In a way, she has. As the police listen in puzzlement to recordings of the mystery caller in which there is only silence, Mortimer reveals that the caller was Death itself. This is the reason that the recipients of the call were all old and had different accounts of how the voice sounded: each will confront Death in his or her own way. Jean Taylor had solved the riddle also. Now able to leave the hospital, she is reunited with Charmian in the nursing home; when informed that there is a phone-call for her from someone who did not leave his name, she tells the nurse that at the moment she is too busy to answer.