In 1947, John Nash (Crowe) arrives at Princeton University. He is co-recipient, with Martin Hansen (Lucas), of the prestigious Carnegie Scholarship for mathematics. At a reception, he meets a group of other promising math and science graduate students, Richard Sol (Goldberg), Ainsley (Jason Gray-Stanford), and Bender (Rapp). He also meets his roommate Charles Herman (Bettany), a literature student.
Les parents de Katherine Coleman apprennent que leur fille possède un don pour les sciences, qu'elle doit parfaire en fréquentant une école pour Noirs (les écoles étant séparées selon la couleur de peau à cette époque) car elle est dotée d'aptitudes intellectuelles supérieures. En 1962, devenue mathématicienne, Katherine Coleman (désormais Goble) travaille au sein d'un groupe de calculatrices humaines, sur le campus ouest du Centre de recherche Langley à Hampton en Virginie aux États-Unis, avec deux amies et collègues : l'aspirante ingénieure Mary Jackson et la superviseure d'équipe Dorothy Vaughan. À la suite du lancement réussi de Spoutnik 1, Al Harrison, directeur du Space Task Group, exige encore plus d'efforts de la part des employés sous sa supervision de crainte que les Soviétiques ne mettent en orbite une bombe H pouvant exploser au-dessus du sol américain. Goble est envoyée dans son groupe pour vérifier les calculs, devenant la première Afro-Américaine à participer à ce groupe de recherche. Au début, elle est ignorée de ses collègues blancs et doit travailler à partir de données partiellement masquées (Goble n'ayant pas le niveau d'habilitation nécessaire) sous les ordres de Paul Stafford, un homme croyant en sa supériorité intellectuelle et refusant de reconnaître le travail de Katherine.
Twenty-year-old Will Hunting (Damon) of South Boston is a self-taught, genius-level intellect with an eidetic memory, though he works simply as a janitor at the Massachusetts Institute of Technology, and spends his free time drinking with his friends, Chuckie (Affleck), Billy (Cole Hauser) and Morgan (Casey Affleck). When Professor Gerald Lambeau (Skarsgård) posts a difficult problem taken from algebraic graph theory as a challenge for his graduate students, Will solves the problem anonymously, stunning both the graduate students and Lambeau himself. As a challenge to the unknown genius, Lambeau posts an even more difficult problem and chances upon Will solving it. Fearing he will lose his sole means of (a meager) income, Will flees and skips going into work the next day. That night, Will meets Skylar (Driver), a British orphan about to graduate from Harvard, who plans on attending medical school at Stanford.
In 1963, Cambridge University astrophysics student Stephen Hawking (Eddie Redmayne) begins a relationship with literature student Jane Wilde (Felicity Jones). Although Stephen excels at mathematics and physics, his friends and professors are concerned over his lack of thesis topic. After Stephen and his professor Dennis Sciama (David Thewlis) attend a lecture on black holes, Stephen speculates that black holes may have been part of the creation of the universe and decides to write his thesis on time.
Growing up, Evan Treborn, and his friends, siblings Kayleigh and Tommy Miller, and Lenny, suffered many severe psychological traumas that frequently caused him to black out. These traumas include being coerced to take part in child pornography by Kayleigh and Tommy's father, George Miller (Eric Stoltz), being nearly strangled to death by his institutionalized father, Jason Treborn (Callum Keith Rennie), who is then killed in front of him by guards; accidentally killing a mother and her infant daughter while playing with dynamite with his friends; and seeing his dog being burned alive by Tommy.
Massachusetts Institute of Technology (MIT) senior math major Ben Campbell (Jim Sturgess) is accepted into Harvard Medical School, but cannot afford the $300,000 fee. Despite a 44 MCAT score and top grades, Ben faces a fierce competition for the prestigious Robinson Scholarship that would pay entirely for the medical school. The director tells him that the scholarship would go to the student who would "dazzle" him.
In 391 AD, Alexandria is part of the Roman Empire, and Greek philosopher Hypatia (Rachel Weisz) is a teacher at the Platonic school, where future leaders are educated. Hypatia is the daughter of Theon (Michael Lonsdale), the director of the Musaeum of Alexandria. Hypatia, her father's slave, Davus (Max Minghella), and two of her pupils, Orestes (Oscar Isaac) and Synesius (Rupert Evans), are immersed in the changing political and social landscape. She rejects Orestes's love (she offers him her bloody menstrual towel, to show him that love has its drawbacks, while studying has none): she prefers to devote herself to science. Davus assists Hypatia in her classes and is interested in science, and is also secretly in love with her.
Jaime Escalante becomes a math teacher at James A. Garfield High School in Eastern Los Angeles. The school is full of hispanic students from working class families who are way below their grade level in terms of academic skills and have a lot of social problems. Escalante seeks to change the school culture to help the students excel in academics. He soon realizes the untapped potential of his class and sets a goal of having the students taking AP Calculus by their senior year. Escalante instructs his class under the philosophy of "ganas", roughly translating into "desire" or "motivation". The students begin taking summer classes in advanced mathematics with Escalante having to withstand the cynicism of other faculty, who feel the students are not capable enough. As the students struggle with the lower expectations they face in society, Escalante helps them overcome the adversity and pass the AP Calculus exams. To his dismay, the Educational Testing Service questions the success of the students, insisting there is too much overlap in their errors and suggests the students cheated. Escalante defends his students, feeling that the allegations are based more on racial and economic perceptions. He offers to have the students retake the test months later and the students succeed in passing the test again despite only having a day to prepare, dispelling the concerns of cheating.
The film opens with the 27-year-old protagonist, young mathematician Catherine (Gwyneth Paltrow) talking to her father Robert (Anthony Hopkins) after he startles her watching TV in the middle of the night. He gives her a bottle of champagne for her birthday, and they chat for a while about the nature of insanity, ending with the revelation that Robert died last week and his funeral is tomorrow.
Max Cohen is the story's protagonist and unreliable narrator. Unemployed, and living in a dreary Chinatown apartment in New York City, Max is a number theorist who believes that everything in nature can be understood through numbers. He is capable of doing simple arithmetic calculations involving large numbers in his head, a skill that impresses Jenna, a small Chinese-American girl with a calculator who lives in his apartment building. Max also suffers from cluster headaches, as well as extreme paranoia, hallucinations, and social anxiety disorder. Other than Devi, a young woman living next door who sometimes speaks to him, Max's only social interaction is with Sol Robeson, his old mathematics mentor who is now an invalid.