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.
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 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.....
Millionaire business man Benedict Farley, a well known eccentric, consults Poirot about bad dreams he has been having. Mr. Farley has dreamed repeatedly of killing himself. Sadly, he does not follow Poirot's advice. When Farley is found dead, Poirot fails to believe that it is suicide and suspects a murder plot relying on hypnosis.
Lovely young actress Valerie St Clair, engaged to marry Prince Paul of Maurania, is the only witness to the murder of Henry Reedburn, a shady and overbearing producer. Prince Paul appeals to Poirot to save Valerie from police suspicion, and the great detective finds he has a conundrum to ponder on, 'When is a murder not a murder?'
As the MQ staff works from home, Poppy struggles with solitude while Brad and David start a charitable competition.
Lord Mayfield, an aircraft manufacturer, tries to trap a spy at his country house, but his plan goes terribly wrong. Lady Mayfield sends for Poirot, who tells a few home truths.
Poirot and Hastings are on board a ship steaming towards Alexandria. Poirot recovers from his usual sea-sickness to take an interest in some fellow passengers, especially the Clappertons. On arrival at Alexandria, Mrs Clapperton is found murdered in her cabin.
Poirot is on holiday at the Palace Hotel, Rhodes, when an Englishwoman, Mrs Valentine Chantry, is poisoned. Poirot unravels several complicated relationships between those staying at the Palace Hotel.
Poirot discovers a murder in his own block of flats, and goes on the trail of boisterous neighbours and strange nocturnal noises in the lift shaft.