Convenience store robber Herbert I. "Hi" McDunnough (Nicolas Cage) and police officer Edwina "Ed" (Holly Hunter) meet after she takes the mugshots of the recidivist. With continued visits, Hi learns that Ed's fianc  has left her. Hi proposes to her after his latest release from prison, and the two get married. They move into a desert mobile home, and Hi gets a job in a machine shop. They want to have children but Ed is infertile, and they cannot adopt because of Hi's criminal record, despite the fact that Ed is a police officer. Devastated, Ed resigns her job. The couple learns of the "Arizona Quints," sons of locally famous furniture magnate Nathan Arizona (Trey Wilson); Hi and Ed kidnap one of the five babies, whom they believe to be Nathan Junior.
Hi and Ed return home and are soon visited by Hi's cellmates, Gale and Evelle Snoats (John Goodman and William Forsythe), who have just escaped from prison. Under the brothers' influence, Hi is tempted to return to his felonious ways. Their problems get worse when Hi's supervisor, Glen (Sam McMurray), proposes wife swapping and Hi assaults him. That night, Hi decides to steal a package of diapers for the baby, but gets carried away and starts to rob the convenience store. Ed sees this and, furious, drives off without him. Hi is then forced to flee on foot from the convenience store, chased by two police officers and an armed cashier, who attempt to shoot him down, as well as a pack of neighborhood dogs, but he manages to outrun and lose them. Ed eventually picks him up, leading to a tense ride home.
At the McDunnough residence the next day, Glen approaches Hi to fire him, and reveals that he has deduced Junior's identity because of the newspaper article he read about Junior missing, and blackmails Hi, threatening to turn him over to the police unless Glen and Dot get custody of Junior. Gale and Evelle overhear this conversation and turn on Hi, tying him to a chair and taking Junior for themselves. Gale and Evelle leave with plans to rob a "hayseed" bank with Junior in tow. When Ed comes home, she frees Hi and the two arm themselves and set out together to retrieve the child. En route, Ed suggests that they should end their marriage after recovering the boy. Meanwhile, Nathan Arizona Sr. is approached by the bounty hunter Leonard Smalls (Randall "Tex" Cobb) who offers to find the child for $50,000. Nathan Sr. declines the offer, believing that Smalls himself is his son's kidnapper. Smalls decides to recover the child anyway to sell on the black market. He begins tracking Gale and Evelle and learns of their bank robbery plans.
Gale and Evelle rob a bank but leave Junior there as they make their getaway. One of the bank's anti-theft dye canisters explodes in their loot sack, blocking the car's windows and incapacitating them. At the bank, Smalls arrives for Junior just ahead of Ed and Hi. Ed grabs the baby and flees; Hi is able to fend Smalls off for a while, but eventually finds himself at Smalls' mercy when Smalls punches Hi in the face several times. As Smalls throws Hi to the ground and prepares to kill him, Hi holds up his hand to reveal that he has pulled the pin from one of the hand grenades on Smalls' vest. Smalls attempts to get rid of the grenade, but he cannot get it off in time and is blown to pieces when the grenade explodes and sets off all his weapons.
Hi and Ed sneak Junior back into the Arizona home and are confronted by Nathan Sr. After Nathan Sr. learns why they took his son, he understands the couple's predicament and decides not to turn them over to the police. He counsels them: when Hi and Ed say that they are splitting up, he advises them to sleep on it. Hi and Ed go to sleep in the same bed, and Hi has a dream about Gale and Evelle reforming after returning to prison; Glen gets his due from a Polish-American police officer after "telling one Polack joke too many"; and Nathan Jr. gets a football for Christmas from "a kindly couple who wish to remain unknown", later becoming a football star. The dream ends with an elderly couple (implied to be Hi and Ed) together enjoying a holiday visit from a large family of children and grandchildren.
Output: Why can't the couple adopt?