American salesman Alan Clay is sent to sell a holographic teleconferencing system to the Saudi government by overseeing a presentation for the King of Saudi Arabia. Clay was offered the job solely because he once met a nephew of the King. Clay is haunted by a previous job at the Schwinn Bicycle Company, where he oversaw the outsourcing of their production to China, which led to hundreds of layoffs in the US. He is also depressed by a messy and costly divorce, which leaves him unable to afford to send his daughter to college. Oversleeping on his first day in Jeddah, due to jet lag, he misses the shuttle bus to the King's Metropolis of Economy and Trade, a largely unfinished planned city in the desert, where Clay is scheduled to give his sales presentation. He hires a driver, Yousef, who describes being in love with a woman whose wealthy and jealous husband has led Yousef to fear for his life. After arriving, Clay is informed that neither the King nor his direct contact, Karim Al-Ahmed, are there. Clay's team is placed in a tent outside the office building, which lacks food and workable internet access. Clay repeatedly oversleeps and calls Yousef repeatedly to drive him to the development. They become closer during the long drives. At the development, he is repeatedly put off and confined to the tent. One day, he slips inside the building and meets Danish executive Hanne. She is sympathetic to his plight, but cannot help him get in contact with the King or Karim Al-Ahmed. She offers him alcohol disguised as olive oil, which Clay has missed since arriving in Saudi Arabia. That night, Clay gets drunk and tries to cut open a lump on his back. Waking the next day, covered with blood from the cut, he goes to a hospital, where he develops an immediate connection with Zahra, the doctor treating him. She performs a biopsy and asks him to return in a few days. After more days without progress, Clay is invited by Hanne to a party at the Danish consulate, where she unsuccessfully tries to seduce him. The next day, after the tent's air conditioning breaks, Clay becomes upset. He again slips into the office building and finally meets Karim Al-Ahmed. Karim assures Clay he will take care of the team's problems but cannot give him a date for the presentation. He shows Clay a new, totally empty condominium development. Back in Jeddah, Clay has a panic attack at his hotel. Mistaking it for a stroke, he calls Zahra and Yousef. Yousef, arriving shortly after Zahra, notices how close they are and after she leaves, chastises Alan for endangering her by making advances, which Clay vehemently denies. Yousef confesses the jealous husband has threatened him and flees to his home town in the mountains for the weekend to let things cool down. Clay accompanies him, passing through Mecca, a city prohibited to non-Muslims. After returning, Clay learns his lump contains precancerous cells and should be removed the next day. When returning to the development, Clay notices the technical problems have been resolved, and successfully gives the King his presentation. However, soon after, Clay learns the Saudis have made a deal with a Chinese company, which can offer a similar product at a cheaper rate. The next day, the operation begins with an unknown male doctor, but at the last moment, Zahra takes over, to Clay's delight. After the procedure, Clay and Zahra exchange increasingly personal and intimate emails, which culminate in a secret meeting. They talk about their families, and Zahra explains she has children and is going through a messy divorce. They are driven to Zahra's beach house, where they go swimming and have sex. Clay writes to his daughter, telling her the deal was unsuccessful but that he has taken a well-paid job selling condominiums in the unfinished city, which will allow him to pay for college for her, and has found a new positive force in his life with Zahra.

College student Jeffrey Beaumont returns to his suburban hometown of Lumberton, North Carolina, after his father, Tom, has a near-fatal stroke. Walking home from the hospital, Jeffrey cuts through a vacant lot and discovers a severed human ear, which he takes to police detective John Williams. Williams's daughter Sandy tells Jeffrey that the ear somehow relates to a lounge singer named Dorothy Vallens. Intrigued, Jeffrey enters her apartment by posing as a pest exterminator. While there, he steals a spare key while she is distracted by a man in a distinctive yellow sport coat, whom Jeffrey nicknames the "Yellow Man". Jeffrey and Sandy attend Dorothy's nightclub act, in which she sings " Blue Velvet ", and leave early so Jeffrey can break into her apartment. Dorothy returns home and undresses; she finds Jeffrey hiding in a closet and forces him to strip at knifepoint, but he retreats to the closet when Frank Booth, a psychopathic gangster and drug lord, arrives and interrupts their encounter. Frank beats and rapes Dorothy while inhaling gas from a canister, alternating between fits of sobbing and violent rage. After Frank leaves, Jeffrey sneaks away and seeks comfort from Sandy. Surmising that Frank has abducted Dorothy's husband Don, and son Donnie, to force her into sexual slavery, Jeffrey suspects that Frank cut off Don's ear to intimidate her into submission. While continuing to see Sandy, Jeffrey enters into a sadomasochistic relationship with Dorothy, in which she encourages him to hit her. Jeffrey sees Frank attending Dorothy's show and later observes him selling drugs and meeting with the Yellow Man. Jeffrey then sees the Yellow Man meeting with a "well-dressed man". When Frank catches Jeffrey leaving Dorothy's apartment, he abducts them and takes them to the lair of Ben, a criminal associate holding Don and Donnie hostage. Frank permits Dorothy to see her family and forces Jeffrey to watch Ben perform an impromptu lip-sync of Roy Orbison 's " In Dreams ", which moves Frank to tears. Afterwards, he and his gang take Jeffrey and Dorothy on a high-speed joyride to a sawmill yard, where he again attempts to sexually abuse Dorothy. When Jeffrey intervenes and punches him in the face, an enraged Frank and his gang pull him out of the car. Replaying the tape of "In Dreams", Frank smears lipstick on his face and violently kisses Jeffrey. Frank then has Jeffrey restrained and beats him unconscious, while Dorothy pleads for Frank to stop. Jeffrey awakens the next morning, bruised and bloodied. While visiting the police station, Jeffrey discovers that the Yellow Man is Detective Williams's partner Tom Gordon, who has been murdering Frank's rival drug dealers and stealing confiscated narcotics from the evidence room for Frank to sell. After Jeffrey and Sandy declare their love for each other at a party, they are pursued by a car which they assume belongs to Frank; as they arrive at Jeffrey's home, Sandy realizes the driver is her ex-boyfriend, Mike. After Mike threatens to beat Jeffrey for stealing his girlfriend, Dorothy appears on Jeffrey's porch naked, beaten, and confused. Mike backs down as Jeffrey and Sandy whisk Dorothy to Sandy's house to summon medical attention. When Dorothy calls Jeffrey "my secret lover", a distraught Sandy slaps him for cheating on her. Jeffrey asks Sandy to tell her father everything, and Detective Williams then leads a police raid on Frank's headquarters, killing Frank's men. Jeffrey returns alone to Dorothy's apartment, where he discovers Don dead and with his ear missing, and Gordon in a lobotomised state from being shot in the head. As Jeffrey leaves the apartment, the "Well-Dressed Man" arrives, sees Jeffrey in the stairs, and chases him back inside. Jeffrey uses Gordon's walkie-talkie to say he is in the bedroom before hiding in the closet. The "Well-Dressed Man" arrives at the apartment and Jeffrey observes he is actually Frank in disguise. Frank begins searching the apartment, and kills Gordon in the process. Jeffrey sneaks out of the closet and grabs Gordon's gun, which he uses to kill Frank when he opens the door. Moments later, Sandy and Detective Williams arrive. Some time later, Jeffrey and Sandy have continued their relationship, Tom Beaumont has recovered, and Dorothy has been reunited with her son.

The four mathematicians are gathered and meet with a top official of the United States Department of Defense. After some discussion, the group agrees that they must be wary with whom to trust and control their solution. The official offers them a reward of $10 million in exchange for their portion of the algorithm, swaying them by attempting to address their concerns. Only one of the four speaks out against the sale, and in doing so is forced to reveal a dark truth about his portion of the solution. Before they sign a license to the government, however, they wrestle with the ethical consequences of their discovery.

Major Vic "Deak" Deakins and Captain Riley Hale, pilots from Whiteman Air Force Base, are flying a stealth bomber with two B83 nuclear bombs on a secret exercise. Deakins suddenly attacks Hale and ejects him from the plane. He then jettisons the unarmed bombs, reports that Hale has gone rogue, and ejects, leaving the plane to crash in a national park in rural Utah. Colonel Max Wilkins send a USAF team to find the missing warheads, and a " Broken Arrow " situation is declared and sends Agent Giles Prentice to oversee the situation. The team locates the warheads in a canyon, but is ambushed by mercenaries and betrayed by Master Sergeant Kelly, who is in league with Deakins. Deakins arrives with Pritchett, the mercenaries' financier. They plan to blackmail the United States government with the threat of detonating the warhead in a populated area. Hale is arrested by park ranger Terry Carmichael, but he convinces her to help him track down Deakins. Deakins' mercenaries commandeer a search-and-rescue helicopter to kill Hale, but Hale and Carmichael bring it down, forcing Deakins' men to continue in Humvees. Hale and Carmichael carjack the Humvee carrying the warheads, escaping to a nearby abandoned copper mine. They soon discover that Deakins was using the mine as a checkpoint. Hale starts to disable one warhead, but Deakins reveals via radio that he has programmed it so that Hale's attempts to disarm it will activate the bomb. Hale and Carmichael take the armed warhead down the shaft, where the mine is deep enough to contain the nuclear blast. Deakins's team arrives and secures the second warhead. After a gun battle deep in the mine, Deakins shortens the countdown of the armed warhead and sabotages the keypad to prevent any disarm attempts while leaving Hale and Carmichael trapped. They escape via an underground river just before the bomb detonates. The resulting nuclear electromagnetic pulse downs an approaching NEST helicopter, allowing Deakins to escape. Pritchett reveals a lack of confidence in Deakins' plans, so Deakins kills him and drives off with Kelly. Carmichael and Hale track the mercenaries to a motorboat to be used for transporting the warhead down the river. While trying to steal the boat, Carmichael is forced to hide on board, while military forces rescue Hale. The U.S. Government is now aware that Deakins betrayed them as he issued a ransom video. Hale deduces that Deakins intends to use a train to transport the warhead to Denver. Stowing away on the train, Carmichael kills one of the mercenaries and tries to sabotage the warhead, but is caught by Deakins, who arms it. Catching up on a helicopter, Hale saves Carmichael before Deakins can kill her. A gunfight ensues, causing the helicopter to crash, killing Wilkins and his pilot, with most of the mercenaries dying in the aftermath. Deakins has prepared a remote control that can either disarm or detonate the warhead and prepares to leave in his own getaway helicopter. Hale sabotages the helicopter's fuel pump, causing it to explode and leaving Deakins and Kelly stranded with the ticking bomb. Deakins shortens the countdown timer out of spite, however, this does not sit well with Kelly. Hale sneaks up on them and engages in a gun battle with Deakins, but not before sending Kelly off a bridge to his death. Carmichael detaches the section of the train with the bomb, but gets into a shootout with the engineer. The engineer is killed and falls on the train brakes, allowing the detached boxcars to catch up. Deakins still has the remote detonator, so he forces Hale to drop his gun and challenges him to a fight. Hale overpowers Deakins, acquires the remote detonator, disarms the warhead, and leaps out of the train. Carmichael also jumps from the train before the detached boxcars catch up and collide. The hurtling warhead flies towards Deakins, who stands in the way of it, launching him into a stack of oil barrels, causing the train to derail and the barrels to explode, killing him. After the explosion, Hale and Carmichael embrace and formally introduce themselves for the first time.