In Stockholm, Sweden, journalist Mikael Blomkvist (Daniel Craig), co-owner of Millennium magazine, has lost a libel case brought against him by businessman Hans-Erik Wennerström (Ulf Friberg). Lisbeth Salander (Rooney Mara), a brilliant but troubled investigator and hacker, compiles an extensive background check on Blomkvist for business magnate Henrik Vanger (Christopher Plummer), who has a special task for him. In exchange for the promise of damning information about Wennerström, Blomkvist agrees to investigate the disappearance and assumed murder of Henrik's grandniece, Harriet, 40 years ago. After moving to the Vanger family's compound, Blomkvist uncovers a notebook containing a list of names and numbers that no one has been able to decipher.
Salander, who is under state legal guardianship due to diagnosed mental incompetency, is appointed a new guardian, lawyer Nils Bjurman (Yorick van Wageningen), after her previous guardian Holger Palmgren suffers a stroke. Bjurman abuses his authority to extort sexual favors from Salander and violently rapes her, not realizing she has a hidden video camera on her bag. At their next meeting she stuns him with a stun gun, rapes him with a dildo, and marks him as a rapist with a tattoo on his chest and stomach. Threatening to disclose the video recording, she blackmails him into writing a glowing progress report and granting her full control of her money.
Blomkvist's daughter Pernilla (Josefin Asplund) visits him and notes that the numbers from the notebook are Bible references. Blomkvist tells Vanger's lawyer, Dirch Frode (Steven Berkoff), that he needs help with his research, and Frode recommends Salander based on the work she did researching Blomkvist himself. Blomkvist hires Salander to investigate the notebook's content. She uncovers a connection to a series of murders of young women from 1947 through 1967, with the women either being Jewish or having Biblical names; many of the Vangers are known antisemites. During the investigation, Salander and Blomkvist become lovers. Henrik's openly national socialist brother Harald identifies Martin (Stellan Skarsgård), Harriet's brother and operational head of the Vanger empire, and Blomkvist marks Martin as a possible suspect. Salander's research uncovers evidence that Martin and his deceased father, Gottfried, committed the murders.
Blomkvist breaks into Martin's house to look for more clues, but Martin catches him and prepares to kill him. While torturing Blomkvist, Martin brags of having killed women for decades but denies killing Harriet. Salander arrives, subdues Martin and saves Blomkvist. While Salander tends to Blomkvist, Martin flees. Salander, on her motorcycle, pursues Martin in his SUV. He loses control of his vehicle on an icy road and dies when it catches fire. Salander nurses Blomkvist back to health and tells him that she tried to kill her father when she was 12. Blomkvist deduces that Harriet is still alive and her cousin Anita (Joely Richardson) likely knows where she is. He and Salander monitor Anita, waiting for her to contact Harriet. When nothing happens, Blomkvist confronts her, deducing that the woman posing as Anita is Harriet herself. She explains that her father and brother had sexually abused her for years, and that Martin saw her kill their father in self-defense. Her cousin Anita smuggled her out of the island and let her live under her identity. Finally free of her brother, she returns to Sweden and tearfully reunites with Henrik.
As promised, Henrik gives Blomkvist the information on Wennerström, but it proves worthless. Salander hacks into Wennerström's computer and presents Blomkvist with evidence of Wennerström's crimes. Blomkvist publishes an article that ruins Wennerström, who flees the country. Salander hacks into Wennerström's bank accounts and, travelling to Switzerland in disguise, transfers two billion euros to various accounts. Wennerström is found murdered. Salander reveals to her former guardian Holger Palmgren that she is in love with Blomkvist. On her way to give Blomkvist a Christmas present, Salander sees him with his longtime lover and business partner Erika Berger (Robin Wright). Heartbroken, she discards the gift and rides away.

Question: What does Henrick Vanger want Mikael Bomkvist to do?
Answer: Investigate the disappearance and possible death of Harriet.
Alternative answer: To investigate the disappearance of his grandniece Harriet.