Matthew Davenheim, a rich banker, disappears on a walk from his country house to the village post office, and Inspector Japp of Scotland Yard is put onto the case. Poirot, who has developed an interest in conjuring tricks, bets Japp five pounds that he can solve the mystery without leaving his apartment.

Poirot calls to see the veiled lady of the title (Lady Millicent Castle-Vaughn) at her London hotel. Lady Millicent is about to be married, but a mean blackmailer called Lavington has got hold of a love letter she wrote years before to another man. Lavington wants twenty thousand pounds, and Poirot makes up his mind to recover the letter himself, even if it means impersonating a Swiss locksmith.

Poirot tells a surprised Hastings he's decided to retire and suggests they take a holiday in the Lake District. On the way, Hastings befriends a young woman who is missing a collection of valuable miniatures. The case involves a famous writer and a beautiful woman in a sports car, and to the relief of Hastings Poirot's thoughts of retirement soon seem far away.

Poirot and Hastings take a holiday in Cornwall, looking for peace and quiet. They meet Miss Nick Buckley, who lives near their hotel at End House, and she tells Poirot she has had three near-fatal accidents in a few weeks. He then finds a bullet hole in her hat, returns with her to End House and is drawn into a complex murder investigation.....

Poirot travels to Cornwall, summoned by a Mrs Pengelley, who suspects her dentist husband is poisoning her because he's fallen in love with his young assistant. Unhappily, Poirot finds Mrs Pengelley dead on his arrival, so he sets out to catch her killer.

Poirot is called in when the British Prime Minister and his secretary are kidnapped in France on their way to a League of Nations conference - and he is given just a day and a half to solve the case.

Marie Marvelle, a Belgian film star, receives threatening letters from a chinaman which demand her famous diamond, the Western Star. Marie visits her old friend Hercule Poirot in London, seeking his help, and Poirot tackles the case with relish.

Hastings and Poirot are playing Monopoly when they are interrupted by Lord Pearson, the elegant Chairman of the London and Shanghai Bank. The banker asks them to investigate the disappearance of a Mr Ling - a highly-valued customer who before he went missing had arranged to sell Lord Pearson a map showing the location of a long-lost silver mine. Ling is found murdered, a young stockbroker is implicated, and Poirot has to untangle a complicated plot.

A glamorous nightclub singer turned spy has stolen the US Navy's plans for a new submarine, and an FBI agent called Burt is in London to catch her. Burt dismisses Poirot as a gumshoe, and Poirot is likewise not impressed by the self-important G-man. Poirot is drawn into the case when it appears that the submarine plans will fall into the hands of the Fascist government in Italy unless they are recovered. When Poirot meets a young couple who've got an amazing bargain on a prime apartment, he sees a connection with the spy case.