A young English woman named Nikky Ferris (Hayley Mills) takes a trip with her folk musicologist aunt, Frances (Joan Greenwood), to a small coastal inn on the Greek island of Crete.
Owner Sophia (Irene Papas) refuses to allow them to stay at her inn, The Moon-Spinners, but Aunt Frances and Sophia's teenage son Alexis (Michael Davis) persuade her into changing her mind. Whilst Nikky and Aunt Frances are in their room, Sophia's brother Stratos (Eli Wallach) demands to know why they chose to stay at his sister's inn and says they should leave, but Aunt Frances insists on staying. Stratos reluctantly agrees to allow them to stay for one night.
During a wedding party at the inn later that evening, Nikky meets a stranger named Mark (Peter McEnery), who invites her and Aunt Frances to have a meal with him. They accept. Their dinner meeting attracts Stratos's suspicious stare, which Nikky notices and points out to Mark. Mark hints there's more to Stratos than appears. At end of the evening, Mark suggests that he and Nikky could meet in the morning to go for a swim in the Bay of Dolphins. Nikky agrees. She comes downstairs the next morning, and quickly learns that Mark has checked out of the inn.
During a walk on the island, she stumbles across Mark who's been shot.
And so her adventure begins, which will lead her to a mysterious wealthy woman, played by former silent film star Pola Negri in her final film role.