Dr. Jim Tanner has always been an expert on human endurance. As the film opens, he is the chairman of a committee of scientists, based at the laboratory of the Space Research Commission in San Marino, CA, tasked to find the limits of human endurance, and those factors that could best enable a human being, say a prospective space pilot, to survive extreme stress and to make instant decisions that could save his life in an emergency. But never before has Tanner found himself pushed to the limits of his own endurance.
On a day like any other, Mr. Arthur Nordlund, US Naval liaison, visits his laboratory and observes some of Tanner's work, as four test subjects are undergoing endurance tests of direct neural stimulation, immersion in freezing-cold water, extreme heat, and high acceleration in a centrifuge. Though impressed, Nordlund is skeptical of the expense, though Tanner seeks to assure him that the work is necessary to the continued viability of the US space program. Biologist Talbot Scott interrupts Tanner, asks to talk with him privately, and warns him that another of their colleagues, anthropologist Henry Hallston, is scheduled to present his report on his human-intelligence survey, and is somehow convinced that all the committee members are "being watched." Scott is afraid that Hallston will embarrass the Committee in front of Nordlund, who is scheduled to attend.
Tanner cannot put off Nordlund, so when the Committee sits, Tanner tries to put off Hallston instead. But Hallston will not be put off. His survey, in which the Committee members participated, reveals that one member--unidentified because the questionnaires were coded--is an unmeasurable genius, so much so that his genius might reward him with preternatural powers that others would be helpless to resist. The other members are openly skeptical, and Tanner despairs of the impression that Hallston is making in front of Nordlund, but Nordlund wants to hear more--and the laboratory's chief administrator, Norman E. Van Zandt, insists on allowing Hallston to have his say, if only to falsify his seemingly outrageous theories. But when physicist Carl Melniker sets up a test of telekinesis by placing a sheet of notepaper on an improvised spindle, and Hallston insists that everyone in the room try together to make the paper spin without touching it, the incredible happens: the paper does spin, slowly at first, then faster and faster.
Tanner is inclined to ignore the whole thing, but his colleague and paramour, geneticist Margery Lansing, is not so sure. She asserts that such a genius could be born today. Then the telephone interrupts them, and at the other end is a frantic Sally Hallston who tells Tanner that her husband had gone to the lab to pick up his questionnaires and had not returned home.
(In fact, Hallston has gone to his office, but never made it out. Someone--presumably the unidentified genius--has impressed on Hallston's mind that the knee-level swinging door has become a gate taller than he, and then a solid wall. After that, Hallston has suffered a heart attack and collapsed at his desk.)
Reluctantly, Tanner drives back to the lab, with Marge in tow. Together they look over Hallston's office, which is unusual in appearance except for some signs that someone has fallen over the desk and spilled some papers. Among the papers, Tanner finds a memo sheet with the name "Adam Hart" scrawled on it. As he is trying to figure out who that person might be, they overhear the centrifuge starting up. Tanner and Marge rush to the centrifuge control room but cannot stop the centrifuge. Tanner rushes down the hallway to pull the main switch, and together he and Marge wait for the centrifuge to stop, and then open it. There they find Hallston, dead, his eyes and tongue still protruding from the high-acceleration forces he has suffered.
The next day, a police detective oversees the removal of Hallston's body and interviews Tanner. That is when things begin to go seriously wrong for Tanner. First, the centrifuge's controls now work properly, contrary to Tanner's story (and experience). Then, the detective and Dr. Van Zandt both accuse Tanner of academic fraud, after Tanner's degree-granting colleges and graduate schools disavow any memory of his having ever enrolled, much less graduated. Then Talbot Scott abruptly refuses to have anything to say to Tanner, this although they had had a friendly conversation only five minutes before. Forced to resign his position, Tanner leaves the laboratory, but not before the detective hints to him that he is now a "person of interest" in what is now a murder investigation.
Unable to figure out what to do next, Tanner wanders aimlessly in the tourist section of San Marino, only to have more weird experiences. A water-dipping bird on display abruptly straightens up, winks at him, then drinks from the water glass and spits at him. In another display window, a company of toy soldiers march smartly into formation, unlimber their muskets, aim at him, and fire! Then he tries to cross the street, but the pedestrian-traffic signal changes its message from DON'T WALK to DON'T RUN. He runs from the intersection and into a fun house, where he finds himself on an empty carousel (with a horse that winks at him) that begins to spin as fast as the centrifuge, and the carousel operator has to pull a main switch of his own to stop it and get Tanner off.
By now, Tanner has had it. He realizes that someone who was at that Committee meeting has set him up. That person, presumably the mysterious Adam Hart, first murdered Hallston, then framed Tanner with the murder, somehow erased Tanner's academic record, and just now tried to kill him--but for a reason that escapes Tanner, failed. Tanner does not want to wait for another murder attempt, but decides to track Adam Hart down himself.
He goes to Joshua Flats, CA, Hallston's hometown, and starts asking questions about Hart, first of a local cafe waitress (who seems unusually romantically interested in Tanner, a thing that also happened to him in the tourist section, but to which he paid no attention), and then of Hallston's parents. The witnesses variously describe Hart, who had been Hallston's schoolmate, as a genius and a sexual magnet. Then the local fill-up station operator offers to take Tanner to Hart's old hangout--but instead dumps Tanner in the middle of the desert next to an Air Force firing range. Tanner is nearly killed again when a squadron of Air Force jets starts strafing the range. He escapes only because he has the presence of mind to light a fire in the brush on the range, prompting the pilots to break off. Furious, he returns to the fill-up station, bursts into the operator's bedroom, and interrogates the operator, who confesses that Adam Hart gave him a standing order ten years earlier to kill any person who dared make inquiries about him. That this operator would blindly obey such a dangerous order after all these years only convinces Tanner further that he is dealing with an assassin with the very preternatural abilities that Hallston most feared. Now he knows the reason for Hallston's attitude: Hallston, having grown up with Hart, had felt instinctively that Hart was a very dangerous person. But Hallston is dead, and Tanner is now Hart's target.
Tanner next goes to see Hallston's widow, who is no help. She now doesn't remember the name Adam Hart (though she had said that Hallston had mentioned him once), doesn't remember what her late husband looked like, and doesn't even remember Tanner, though for some weird reason she also seems romantically interested in him, though Tanner puts that down to her being drunk. Next he goes to the home of Margery Lansing, but is violently attacked by Carl Melniker, who earlier has invaded Marge's home, bound her, and lain in wait for Tanner, thinking him to be the unknown genius who had killed Hallston. Tanner manages to subdue Melniker long enough to convince him that he means him no harm, and unbinds Marge. The three scientists then try to have a rational discussion about what they're facing, and how to combat it--or survive it. The three decide that for now they don't dare fall asleep, so they drive to a hotel in nearby Santa Lisa, that his hosting a salesmen's convention, and in effect, crash the party. While there, Tanner sees a newspaper with several copies bearing a partial headline: DON'T RUN! This turns out to be part of a legitimate headline--but why the newspaper would run a banner headline about the city's mayor telling an opposing candidate not to run against him seems most strange. Later, despite their best efforts, Melniker winds up dead, and Tanner and Marge have to flee.
Tanner and Marge next drive to the apartment complex where Arthur Nordlund lives. As they drive into the parking garage, they notice Nordlund abruptly suffering an apparent heart attack and then stumbling into an elevator. The elevator starts to go up, then starts back down, then gets stuck between floors. Tanner climbs to the top floor, enters the elevator shaft, and tries to climb down to the elevator by rapelling down the cable--but then the elevator starts to go up, threatening to crush Tanner. Tanner quickly lets himself down to the elevator cab, opens the trap door, enters the cab, and finds Nordlund, who is near death.
Now that Hallston and Melniker are dead, Nordlund seems to have been attacked himself, and Tanner trusts Marge implicitly, Tanner now narrows his list of suspects to Van Zandt and Scott. Nordlund, who has now joined their group, urges a homicidal counterstrike against the unknown genius, but Tanner does not wish to kill a man indiscriminately, not without knowing which man is the right man. Tanner then visits Van Zandt's home, but succeeds only in speaking to Mrs. Van Zandt before realizing that Dr.Van Zandt suspects Tanner of murder. Then Tanner barely escapes alive when someone (not Norman Van Zandt) tries to run him down. He drives away, only to find himself in an out-of-control car with brakes that don't hold. He splashes into a river and escapes by unfastening the car's convertible top, and then is picked up by police. They take him to a hospital, where the detective first tells him that he is under arrest--for the murders of Dr. and Mrs. Van Zandt, who have died when their house burned down shortly after his visit to it. The detective also informs him that Nordlund has disappeared.
Then comes word that Talbot Scott has locked himself in the auditorium at the laboratory, is armed, and has demanded to see Tanner. Tanner insists that he can talk to Scott and reason with him, and reluctantly the detective lets Tanner go to the laboratory. There he listens incredulously as Scott babbles disconnectedly about Tanner being the genius, and Scott's willingness to serve Tanner however he wishes if only Tanner will not kill him. In the end, the out-of-control Scott dies in a gun battle with police.
At last Tanner realizes who is responsible for his set-up and the murders of his colleagues: Arthur Nordlund, otherwise known as Adam Hart. Instead of running, Tanner walks the corridor and looks at some notes that someone had left, concerning the effect of intense pain on the heart. Marge then reveals herself, telling Tanner that she is glad that he has stopped running and will now stand and fight--because she, too, realizes that Nordlund is responsible.
Tanner confronts Nordlund directly in the corridor of the laboratory. Nordlund then unleashes his full telepathic fury against Tanner, seeking to hypnotize him into believing himself plunged into freezing cold, then searing heat, and finally into the unforgiving void of outer space. Somehow Tanner survives, though he physically collapses--but does not black out. Now he knows why Nordlund/Hart wanted him dead: there are in fact not one genius, but two: Adam Hart and himself! Now he fights back directly, and kills Nordlund by telekinetically squeezing his heart and holding it compressed.
Marge tells Tanner, in wonderment, that she ought to have suspected earlier: Adam Hart would never have revealed himself by making the paper spin at the original meeting, and everyone else (except Van Zandt) had tried individually to spin the paper and failed, so therefore Tanner had been the genius all along. Nordlund had tried to kill Tanner after deciding that the world wasn't big enough for two such geniuses, and had eliminated his colleagues to dispose of all possible witnesses (except for Marge, perhaps because he hadn't had a chance to eliminate her yet). Now that Tanner realizes the kind of genius that he is, he and Marge leave the laboratory, with Tanner speculating idly on whether his newfound power will corrupt him--and unable to answer his own question.