Juliet asks Romeo not to swear by the moon because the moon is always changing its shape and position. Therefore, a promise sworn on the moon could also be prone to changing. Her request is part of the famous balcony scene, which is the second scene in Act II of the play "Romeo and Juliet."
Romeo attempts to profess his love to Juliet in this scene, but Juliet cuts him off mid-sentence by saying: "O, swear not by the moon, the inconstant moon; that monthly changes in her circled orb; lest that thy love prove likewise variable." Based on these words, Juliet believes that a promise sworn on the moon is just as changeable as the moon itself. Romeo then asks what he should swear by, and Juliet counsels him not to swear by anything at all except for Romeo's "gracious self." As Romeo begins to respond to her words, Juliet cuts Romeo off again, and she asks him not to swear anything at all. She does not want them to pledge themselves to each other too rashly. She then asks him to send a message to her the next day, and in that letter, he should ask for marriage.