In Dresden, Germany, in 1839, pianist virtuoso Clara Wieck performs at a command performance of the King.  After completing a piano concerto by noted composer Franz Liszt, Clara introduces "Traumerei," a piece composed by her lover, an unknown composer named Robert Schumann.  Backstage, Clara is chastised by her overbearing father and teacher, Professor Wieck, for playing the composition, but Clara replies that she is in love with Schumann.

When Schumann asks for the underage Clara's hand in marriage, Wieck forbids the union, declaring that Schumann is unworthy of his famous daughter.  Defying her father, Clara petitions the court for permission to marry.  Just as the judge is about to rule in her father's favor, Liszt defends the marriage and praises Schumann's talent, thus winning the court's consent.  As a wedding gift, Schumann writes a composition for his bride titled "Dedication."

Years later, Schumann has failed to attain recognition as a composer and is forced to support his family by giving music lessons.  Overburdened with seven young children, Clara has stopped playing the piano to rear her brood.  On New Year's Eve, Johannes Brahms, a promising young composer, comes to study with Schumann.  Although Brahms is instantly smitten by Clara, Schumann invites him to move into the family's home.

At their New Year's party, the family and their guests follow tradition by telling their fortunes with melted tin soldiers.  When Schumann's molten glob of metal congeals into the shape of a coffin, the room falls silent.  Soon, Brahms becomes an indispensable member of the family, helping with the chores and the children and even convincing the housekeeper, Bertha, who has quit in frustration, to return to work.

Strain and overwork start to affect Schumann's mind, though, and he begins to suffer chronic headaches.  To help with the family's finances, Clara briefly returns to the concert stage, causing Schumann increased feelings of failure and unworthiness.  While Schumann throws himself into completing his opera Faust, Clara declines a lucrative offer to continue on the concert stage.  Slipping further into depression, Schumann begins to compose morose, disturbing music, prompting Clara to worry that he may suffer the same fate as his sister, who committed suicide after suffering a nervous breakdown.

When Schumann's opera is rejected, Clara hides the bad news from her husband and Brahms turns to Liszt for help.  Interceding on his friend's behalf, Liszt convinces Reinecke, the owner of a concert hall, to open his season with Faust.  No longer able to bear his unrequited love for Clara, Brahms leaves the Schumann home.

While conducting the debut of his opera, Schumann suffers a mental breakdown and is subsequently committed to a mental institution.  When, during a visit from Clara, Schumann offers to play his "new" composition, "Traumerei," Clara knows all is lost; Schumann dies shortly thereafter.

Five years later, Brahms, now having achieved fame and fortune, comes to visit Clara.  Surprised to find Clara still in mourning and having forsaken the piano, Brahms lectures her about her late husband's determination.  Resentful, Clara rejects Brahms invitation to join him in Cologne for the debut of his new symphony, but later changes her mind.  In Cologne, Brahms is so thrilled to glance up and see Clara in the audience that he whisks her out of the hall in the middle of the concert.  Taking her to a wine garden, Brahms proposes to Clara, but when a gypsy violinist plays the strains of "Dedication," Clara realizes that she is still in love with her husband and must preserve his legacy by playing his music throughout the world.  Returning to the stage in Dresden, she reprises "Traumerei" for the King.