The original Broadway production, starring Mary Martin and Theodore Bikel, opened in November 1959, and the show has enjoyed numerous productions and revivals since then. It has also been made into an Academy Award-winning 1965 film musical starring Julie Andrews and Christopher Plummer. The Sound of Music was the final musical written by Rodgers and Hammerstein; Hammerstein died of cancer nine months after the Broadway premiere.

After viewing The Trapp Family, a 1956 Austrian film about the von Trapp family, and its 1958 sequel, The Trapp Family in America (Die Trapp-Familie in Amerika), stage director Vincent J. Donehue thought that the project would be perfect for his friend Mary Martin; Broadway producers Leland Hayward and Richard Halliday (Martin's husband) agreed. The producers originally envisioned a non-musical play that would be written by Lindsay and Crouse and that would feature songs from the repertoire of the Trapp Family Singers. Then they decided to add an original song or two, perhaps by Rodgers and Hammerstein. But it was soon agreed that the project should feature all new songs and be a musical rather than a play.

Details of the history of the von Trapp family were altered for the musical. Georg Ludwig von Trapp lived with his family in a villa in Aigen, a suburb of Salzburg. The real Maria von Trapp was sent to be a tutor to one of the children, not a governess to all of them. The Captain's oldest child was a boy, not a girl, and the names of the children were changed (at least partly to avoid confusion: the Captain's second eldest daughter, the third of the seven, was also called Maria). The von Trapps spent some years in Austria after Maria and the Captain married – they did not leave Austria right away. They did not flee Austria; they merely boarded the train to Italy, not Switzerland. Maria von Trapp is said to have been unhappy with the movie's portrayal of her husband as having been cold and stern prior to her arrival, which she and their children strongly dispute. It is odd that Maria would make such a criticism, because the whistle scene near the beginning of the movie is taken from her 1949 book, "The Story of the Trapp Family Singers,", and that book has a chapter entitled, "The Baron Doesn't Want It,", describing the activities in which the children could not engage for fear of offending their father, the captain. In 1972, however, Maria wrote another book, "Maria," in which she states that the captain was not home when she first met the children.

In Salzburg, Austria, just before World War II, nuns from Nonnberg Abbey are singing the Dixit Dominus. One of the postulants, Maria Rainer, is missing. On the mountainside near the abbey, Maria expresses her regret to leave the beautiful hills ("The Sound of Music"). She returns to the abbey after the gates are locked; the next day, the Mother Abbess and some of the other nuns consider what to do about her ("Maria"). Maria explains that she was raised on that mountain and apologizes for singing in the abbey garden without permission. The Mother Abbess joins her in song ("My Favorite Things"), but later tells Maria that she should spend some time outside the abbey to help her decide whether she is ready for the monastic life. The seven children of widower Captain Georg von Trapp need a governess, and Maria will act as their governess until September.

At his villa, von Trapp, a decorated World War I Captain of the Austro-Hungarian Navy, informs Franz, the butler, and Frau Schmidt, the housekeeper, that a new governess is coming. He also instructs them to prepare for his return from Vienna with two guests. Maria arrives, and the Captain explains her duties. He then summons the children with a bosun's whistle, and they march in, clad in uniforms. He introduces the children (Liesl, Friedrich, Louisa, Kurt, Brigitta, Marta, and Gretl) and teaches her their individual signals; but she openly disapproves of this militaristic approach. When alone with them, she breaks through their wariness, and after learning that they do not know how to sing, she teaches them the basics of music ("Do-Re-Mi").

That evening, Rolf, a young messenger, delivers a birthday telegram to Franz and then meets with Liesl outside the villa. He claims he knows what is right for her because he is a year older than she is ("Sixteen Going on Seventeen"). They kiss, and Rolf runs off.

As Maria prepares for bed, Frau Schmidt gives her material to make new clothes, as she had given all her worldly possessions to the abbey for the poor. Maria asks for more material to make play clothes for the children, but Frau Schmidt refuses on grounds that they "march, not play". As Maria says her evening prayers, Liesl slips through the window, soaking wet from the thunderstorm. Maria agrees to keep her secret. The other children run in, frightened by the storm. To comfort and cheer them, Maria sings "The Lonely Goatherd".

Captain von Trapp arrives a month later with Baroness Elsa Schräder and Max Detweiler, and they wonder why the children are not there to greet them. When the Captain goes to look for the children, Elsa tells Max that something is preventing the Captain from marrying her. Max opines that only poor people have the time for great romances ("How Can Love Survive"). Rolf enters, looking for Liesl. Surprised by the Captain, he greets them with "Heil". The Captain orders him off the property, maintaining that he is Austrian, not German. Maria and the children leapfrog in, wearing play-clothes made from the old drapes in Maria's room. Infuriated, the Captain sends them off to clean up and change. Maria firmly tells him that the children need him to love them, and he angrily orders her back to the abbey. As she apologizes, they hear the children singing "The Sound of Music" (Reprise), which Maria had taught them, to Baroness Schräder. The Captain joins in, and at the end he embraces the children. Alone with Maria, he asks her to stay, thanking her for bringing music back into his house. Elsa is suspicious of Maria until Maria explains that she will be returning to the abbey in September.

The Captain gives a party to introduce Elsa to his friends, and some of the guests argue over the Anschluss. Kurt asks Maria to teach him to dance the Laendler. She demurs, but he insists, and she attempts it. When he is unable to negotiate a complicated figure, the Captain steps in to demonstrate. Maria and the Captain dance until they come face-to-face, and Maria breaks away, embarrassed and confused. When Max arrives at the party, the Captain realizes that he needs another woman to balance the dinner table and asks Maria to fill this role. Maria and Brigitta discuss the expected marriage between Elsa and the Captain, and Brigitta tells Maria that she and the Captain are in love with each other. Elsa asks the Captain to let the children say goodnight to the guests with a song. The Captain resists; but Elsa nevertheless starts them off singing "So Long, Farewell". Max is amazed at their talent and decides that he needs them for the Saltzberg Festival, which he is organizing. After the guests leave for the dining room, Maria unhappily slips out the front door with her luggage.

At the abbey, Maria tells the Mother Abbess that she is ready to take her monastic vows; but the Mother Abbess realizes that Maria is running away from her feelings. She tells Maria that she must return to face the Captain and discover if they love each other, and that, by actively searching for it, Maria must find the life she was meant to live ("Climb Ev'ry Mountain").