A young man wanders through a heavy rainstorm, finding his way into a nearby restaurant. The restaurant's employees try to determine if he needs help. Despite his manic mode of speech being difficult to understand, a waitress, Sylvia, learns that his name is David Helfgott and that he is staying at a local hotel. Sylvia returns him to the hotel, and despite his attempts to engage her with his musical knowledge and ownership of various musical scores, she leaves.

As a child, David is growing up in suburban Adelaide, South Australia and competing in the musical competition of a local Eisteddfod. Helfgott has been taught to play by his father, Peter, who is obsessed with winning and has no tolerance for imperfection, dishonour and disobedience. While playing at the Eisteddfod, David is noticed by Mr. Rosen, a local piano teacher who, after initial resistance from Peter, takes over David's musical instruction.

As a teenager, David wins the state musical championship and is invited by concert violinist Isaac Stern to study in United States. Plans are made to raise money to send David off to America. Initially, his family is supportive, but then Peter forbids David to leave, thinking his absence would destroy the family. To make matters worse, Peter begins physically and mentally abusing David, which causes strain to the rest of the family.

Crushed, David continues to study and befriends local novelist and co-founder of the Communist Party of Australia, Katharine Susannah Prichard. David's talent grows until he is offered a scholarship to the Royal College of Music in London. This time, David is able to break away from his father with the encouragement of Katharine. However, his father issues an ultimatum, effectively banishing David and saying that David will never return home and never be anybody's son.

In London, David studies under Dr. Cecil Parkes and enters a Concerto competition, choosing to play Sergei Rachmaninoff's enormously demanding 3rd Concerto, a piece he had attempted to learn as a young child to make his father proud.

As David practices, his behavior becomes increasingly unhinged. David wins the competition, but collapses and suffers a mental breakdown. He is then admitted to a psychiatric hospital upon his return to Australia, David receives electric shock therapy to treat his condition. David recovers to the point where he is able to return to Adelaide. However, his attempts to reconcile with his father are rebuffed due to the latter's mindset that David abandoned his family. This causes David to relapse and he is forced to be readmitted to a mental institution.

Years later, a volunteer at the institution recognises David and knows of his musical talent. The woman takes him home but discovers that he is difficult to control, unintentionally destructive, and needs more care than she can offer. As time passes, David has difficulty adjusting to life in broader society again, and often leaves the hotel to stimulate his interests.

The next day, David returns to the restaurant where the patrons are astounded by his ability to play the piano. One of the owners befriends David and looks after him. In return, David plays at the restaurant. Through the owner, David is introduced to an employee named Gillian (Lynn Redgrave). David and Gillian soon fall in love and marry. With Gillian's help and support, David is able to come to terms with his father's death and to stage a well-received comeback concert, presaging his return to professional music.