An American family, Dr. Ben McKenna (James Stewart), his wife Jo (Doris Day) and their son Hank (Christopher Olsen) are on vacation, traveling in Morocco. They befriend a fellow traveler, the mysterious Louis Bernard (Daniel Gelin), on a bus. Bernard is friendly enough, but Mrs. McKenna becomes suspicious and thinks he is hiding something. At the hotel, Bernard offers to take the McKennas out to dinner that night, but suddenly cancels when a sinister-looking man arrives at the door of the McKennas' hotel room. Later, the couple meets another vacationing couple, the Draytons (Bernard Miles and Brenda De Banzie) from England, at dinner in a local restaurant. Louis Bernard also comes to the restaurant, but sits at a separate table from the McKennas and the Draytons.
The next day, while exploring a busy outdoor marketplace in Marrakech, the McKennas see a man being chased by police officers. Shortly afterward, the same man approaches them after being stabbed in the back by an assassin. Ben discovers that the man is wearing brown makeup and is really Louis Bernard. Before dying, Bernard whispers into Ben's ear a terrible secret: that someone's life is in danger. A foreign statesman will be assassinated in London very soon, and to contact the London authorities to try 'Ambrose Chapel'. After the murder, The Draytons offer to return Hank to the hotel while Dr. and Mrs. McKenna are questioned by the authorities. The interrogator reveals that Louis Bernard was a spy working for the French government. At the police station, Ben receives a telephone call from a mysterious man who informs him that Hank has been kidnapped and threatens to harm him unless the McKennas say nothing to the police about Bernard's last words.
Ben and Jo travel to London to investigate where a local police inspector wants them to cooperate with them with what they know about an assassination plot and who may be the victim. But Ben continues to refuse to cooperate out of fear for his kidnapped son's safety.
After following a number of false leads, Ben tracks the kidnappers to a church, called Ambrose Chapel, where Mr. Drayton is posing as the minister. Ben learns that the Draytons are the leaders of an anarchist terrorist group involved in a plot to assassinate a European head of state during a symphony orchestra concert at the Royal Albert Hall.
Ben and Jo separately track the killer to the concert, where he is to shoot the dignitary at the exact moment of time when the orchestra's music features a loud and climactic cymbal crash. At the moment of truth, Jo screams. The sudden unexpected sound causes the assassin to misfire. Ben chases the assassin, who falls to his death from a balcony.
The couple then follow the kidnappers to the ambassador's residence in London, where they are welcomed as heroes for saving the head of state's life. Mrs. Drayton, unable to be complicit in the plan to kill Hank, helps the boy find his father. Mr. Drayton tries to escape with the two as hostages, but is struck by Ben and falls down the stairs to his death when the gun he is holding fires accidentally. It is never explained what ultimately happens to Mrs. Drayton, but she does witness her husband's death.
In the final shot, Ben and Jo, along with their son Hank return to their London hotel room to inform their visiting guests about taking so long for they had to pick up Hank.