After Arthur “Buck” Taylor, son of a deceased police officer, is turned down by the police academy as psychologically unfit for the force, he telephones Todd Arthur and leaves a message on his answering machine, threatening to kill him that night. Todd goes to the police, only to be told they are short on staff and to leave the machine’s cassette tape. Todd returns home, where Buck kills him with a steak knife. The murderer leaves a note reading, “Catch me.” The next day, Sam Dietz arrives at the station as a newly promoted detective. He and his partner, Bill Malloy, are assigned to Todd’s murder case. At the crime scene, Dietz figures out what transpired, but Malloy warns him to wait for laboratory reports before making any conclusions. Meanwhile, at the firing range, Buck remembers how his father, Ike Taylor, took him shooting when he was a small boy, and beat him for missing the target. Buck was also forced to run an obstacle course while his father fired bullets over his head. No matter how well Buck did, Ike would beat him for being too slow. Buck goes home and picks Angela Taylor’s name from the telephone book and prowls on her roof. Hearing a disturbance, she telephones police as he crashes through a skylight. Angela hides in a dryer machine, but Buck finds her and garrotes her with piano wire. Later, Dietz and Malloy find a note at the crime scene that reads, “What’s taking you so long?” Dietz begs Malloy to let him investigate the crimes instead of waiting for laboratory reports. Malloy refuses and Captain Blakely orders Dietz not to talk to the press. Dietz acquires a psychological profile of the killer from police psychologist Doctor Park. The next night, Buck strikes again and kills Arthur Taylor, a movie producer, with a corkscrew. As he leaves, he hears Arthur’s mistress calling him and strangles her. When Dietz and Malloy arrive at the scene, they find another taunting note, and Malloy notices the similarity in the victims’ names. The next morning, Malloy telephones Dietz to report two more murders of people with the name “Arthur Taylor.” The newspapers learn of the multiple murders and label Buck “The Sunset Killer.” It is reported that Dietz and Malloy are the detectives in charge of the investigation. Dietz and Malloy canvas all the “Arthur Taylors” in the telephone book to warn them they may be in danger. In a rundown weekly motel, they meet Buck. Malloy recognizes a picture of Buck’s father, and realizes he knew Buck as a boy. Dietz finds a revolver and suggests Buck keep it close until they catch the serial killer. After they leave, Buck puts his revolver in his mouth, but decides the police must kill him. That afternoon, Buck goes to Malloy's home and shoots the cop to death. Dietz thinks Malloy’s death is unconnected to the serial killer, until he opens his front door and finds a page from the telephone book with Malloy’s name and a message that reads: “If I’m so sick, why can’t you catch me?” He orders his wife Carol to take their son, Corey, to her mother’s house. With Dr. Park’s consent, Dietz leaks a phony psychological profile of the killer to the press, stating that he suffers from impotency. Hoping that he will go after Dr. Park, Dietz cons a police lieutenant into assigning him four men, and they stake out the doctor’s home. When Buck arrives at Park’s house, he spots Dietz’s car and walks away. An officer named Terrell telephones Dietz and plays him the answering machine tape from the first victim, Arthur Todd. Dietz recognizes Buck’s voice. When he and another policeman go to Buck’s hotel, they find a telephone book missing the page with Dietz’s address. Dietz tries to call home, but the telephone is off the hook. Dietz rushes home to find Buck pointing a gun at his family. He screams for him to drop the weapon, but Buck hallucinates that Dietz is his father and raises his pistol. Dietz empties his revolver into the man’s chest. As Buck bleeds to death, he pulls out a note that reads, “Thank you.”