George Thomason (Tom Georgeson) and his right-hand man, Ken Pile (Michael Palin), a beleaguered animal lover with a bad stutter, plan a jewel heist and bring in two Americans to help: an alluring con artist, Wanda Gershwitz (Jamie Lee Curtis) and a "weapons man", Otto West (Kevin Kline). Wanda and Otto are lovers pretending to be siblings so that Wanda can work her charms on George and Ken. Wanda and Otto plan to betray Ken and George after the heist, and vice-versa. Wanda, it later transpires, is also planning to betray Otto.
After the robbery, Wanda and Otto betray George to the police, intending to take all the loot for themselves, but discover that George and Ken have moved the loot to a new location. Wanda decides to seduce George's unhappily married lawyer, Archie Leach (John Cleese) to find out where it is. Meanwhile, Ken is charged by George to silence a key witness, and Otto exerts his energies in fits of jealousy over Wanda and Archie and of impatience with Ken and Britain in general. Otto's interference, and other incidences of bad luck, lead Wanda and Archie's liaisons to go disastrously wrong.
There is subsequent confusion and slapstick. For example, in various attempts to kill the witness, the animal-loving Ken accidentally instead kills her three Yorkshire Terriers one by one. The witness eventually suffers a fatal heart attack when her third Terrier is killed. The title is derived from the name of Ken's favorite tropical fish, which Otto eats alive, to Ken's distress, during the farcical torture scene.
Archie and Wanda gradually fall in love and the movie ends with those two leaving the country with the jewels. Ken finally has his revenge on Otto by running him over with a steamroller (which he somehow survives). Title cards at the end of the movie reveal their futures: "Archie and Wanda were married in Rio, had seventeen children, and founded a leper colony. Ken became Master of Ceremonies at the London Sea World. Otto emigrated to South Africa and became Minister for Justice.
At the London flat of Ken an animal lover with a large tank of tropical fish (the large angelfish is named "Wanda", after his flatmate George's sexy American girlfriend) and a nasty stutter, a meeting takes place with himself, the attractive Wanda, her lover George and her "brother" Otto. They plan to rob the store of a prominent London jeweler and steal a cache of diamonds. Otto, a belligerent and boisterous former CIA operative, will provide the weapons and plot the strategy to rob the business with no deaths involved, at the behest of George, who has brought the idea for the caper to the group.
The robbery goes off without a hitch and as the team makes their getaway, they nearly run over an elderly woman out walking her dogs. She sees George without his ski mask on. The team splits up, George, Wanda and Otto place the diamonds in a small safe housed in a garage and continue with their plan to lay low for a few days before flying out of the country. Just after Wanda and Otto leave the garage, George returns, retrieves the loot and drives off with it.
Back at Otto's flat, Wanda (who reveals that Otto is not her brother but her 2nd lover) fends off Otto's attempts at seducing her long enough to have Otto phone the police and give them George's name.
George is arrested at his own flat and implicated with the crime; he has particles of on the leg of his trousers from the jeweler's. George is allowed to meet with Ken and gives him a small key, which Ken hides in the tiny treasure chest of his aquarium -- Wanda sees Ken hiding the key and later takes it, hiding it in her locket. When Wanda and Otto meet with George, he is immediately suspicious of Otto. At a police lineup, the woman that George nearly killed with his car points him out as the man who nearly killed her dogs. At the grand jury hearing, George gives Ken a small nod, identifying the old lady.
Wanda sees that George is being represented by a barrister named Archie Leach. Wanda begins to play up to him, trying to find out more information about George's case. Archie initially tells her he is not allowed to speak to her since she's been subpoenaed as a character witness for George. Archie, himself in a loveless marriage, later acquiesces, saying that it's permitted for him to talk to Wanda as long as they don't discuss the case itself. The two meet clandestinely several times -- at Archie's house, while his wife and daughter are out for the evening, Wanda loses her locket with the key inside. Archie's wife, having returned with their daughter after they had a tire blow out, believes the locket is a gift to her from Archie. Wanda demands the locket be returned.
Wanda is also faced with Otto's extreme jealousy with regards to Wanda's attempts to seduce Archie. Several times Otto threatens Archie with bodily harm, even dangling him out a window. Wanda is furious, explaining to Otto that his jealous behavior may get them in trouble & they still won't be able to find out where George hid the diamonds. She convinces Otto to apologize to Archie.
Archie's attempts to get the locket back from his wife mostly fail. One day he goes to his own house and stages a robbery. While he loots a few small objects from his wife's jewelry collection (and breaks a small ceramic figurine) Otto, who'd arrived at Archie's house to apologize, finds him and beats him, believing he's helping out Archie by catching a random burglar. When he discovers that he's beaten Archie himself, he leaves him tied up and flees. Archie, just as his wife arrives home and becomes distraught at the minor destruction her husband had staged, hides Wanda's locket in his mouth until his hands are freed and he can spit it out and hide it in his pocket. He rushes off to a flat belonging to a friend and meets Wanda, giving her the locket. The two decide to have sex -- Archie inadvertently finds that Wanda can be easily seduced when he speaks any foreign language, until they're interrupted by a family that had rented the flat. Archie is caught naked and embarrassed. He calls Wanda and says he's going to end their tryst. Right after he does so and returns home, he's intercepted by Otto, who apologizes to him and grants him permission to continue his affair with Wanda. However, Archie's wife, Wendy, is listening from an open window above.
Wanda tries unsuccessfully to identify George's key and Ken, charged by George with killing the elderly woman with the dogs, tries to make her death look accidental. He only succeeds in killing her dogs one by one until, when he accidentally kills the last dog, the woman dies of a heart attack on the street. Ken reports to George, who gives Ken the go-ahead to retrieve the loot & pack for their escape destination: Rio de Janeiro, saying they'll take both Wanda and Otto & eliminate Otto later. At Ken's flat, Otto figures out that Ken knows where the loot is & tortures him: he ties Ken up and eats all the fish in his aquarium one at a time, including the beloved angelfish Wanda until Ken confesses that the safety deposit box is in a hotel near Heathrow Airport. However, the key is not in the aquarium and Otto calls Wanda, who tells him she'll give him the key and the loot if they flee to Rio together after she testifies on George's behalf.
In court, Wanda takes the stand and Archie questions her about her relationship with George. She portrays him as an honest man until she drops a few details that clue the court into George's involvement in the crime, stating that George had left their flat armed with a shotgun right before the crime. Archie, surprised by Wanda's turn, calls her by her first name and calls her "Darling". Wendy, observing in the mezzanine, is silently infuriated.
George becomes enraged at Wanda's comments and attacks her, causing general melee in the room while the guards try to subdue him. Archie receives a knock on the head from George and is left alone in the courtroom. When Wendy approaches him, she slaps him and tells him she's filing for divorce. Archie then takes a private meeting with George, assuring him that he'd arranged to lead Wanda on the witness stand to get George exonerated. He strikes a deal with George to receive a lesser sentence and convinces George to reveal the location of the diamonds: George tells Archie he'll find Ken back at his flat, who knows where they are.
Outside the court building, Archie finds Wanda trying to sneak off and takes her in his car to George's flat. Archie goes in alone to find Ken and possibly face off against Otto. He finds Ken tied up and hears a car collision outside: Otto, having found Wanda alone, takes off with her after the minor fender-bender to the airport. Archie tries to get the nervously stuttering Ken to tell him where the diamonds are hidden. He finally gets Ken to write down the name of the hotel and they take off for the airport.
At the Cathcart, Wanda and Otto retrieve the diamonds and get their tickets to Rio. As they walk to the gate, Wanda clubs Otto and locks him in a small closet. Archie finds Otto lagging behind Wanda after she'd clubbed him and is able to grab Otto's Walther PPK but tricks Archie into dropping it and grabs it himself. Having a few minutes before he has to leave with Wanda, Otto takes great delight in humiliating Archie by making him stand in a barrel of used oil outside the terminal while insulting him and his British heritage. He prepares to shoot Archie when he sees Ken driving toward him on a steamroller. Thinking that Ken could never catch up to him, much less kill him, Otto steps into a recent batch of wet concrete and is stuck. Frantically apologizing to the vengeance-bent Ken for eating his fish, he's run under the steamroller. Ken, overjoyed at gaining revenge on his tormentor, suddenly loses his stutter and is arrested.
Archie is able to make it to Wanda's plane and the two reconcile. As they take off, Otto, still alive, suddenly appears at their window but is blown off. Closing titles tell us that Archie and Wanda married, had children & founded a leper colony. Ken became master of ceremonies at London Sea World. Otto left for South Africa and became a minister of justice.