American single mother Ann Lake (Carol Lynley), who recently moved to London from New York, arrives at the Little People's Garden preschool to collect her daughter, Bunny. The child has mysteriously disappeared. An administrator recalls meeting Ann but claims never to have seen the missing child. Ann and her brother Steven (Keir Dullea) search the school and find a sinister woman living upstairs, who claims she collects children's nightmares. In desperation, the Lakes call the police and Superintendent Newhouse (Laurence Olivier) arrives on the scene. Everyone becomes a suspect and Superintendent Newhouse is steadfast, diligently following every lead. The police and Newhouse decide to visit the Lakes' new residence.