Although the character appeared previously in Barrie's book The Little White Bird, the play and its novelisation contain the story of Peter Pan mythos that is best known. The two versions differ in some details of the story, but have much in common. In both versions Peter makes night-time calls on the Darlings' house in Bloomsbury, listening in on Mrs. Mary Darling's bedtime stories by the open window. One night Peter is spotted and, while trying to escape, he loses his shadow. On returning to claim it, Peter wakes Mary's daughter, Wendy Darling. Wendy succeeds in re-attaching his shadow to him, and Peter learns that she knows lots of bedtime stories. He invites her to Neverland to be a mother to his gang, the Lost Boys, children who were lost in Kensington Gardens. Wendy agrees, and her brothers John and Michael go along.
Their magical flight to Neverland is followed by many adventures. The children are blown out of the air by a cannon and Wendy is nearly killed by the Lost Boy Tootles. Peter and the Lost Boys build a little house for Wendy to live in while she recuperates (a structure that, to this day, is called a Wendy House.) Soon John and Michael adopt the ways of the Lost Boys.
Peter welcomes Wendy to his underground home, and she immediately assumes the role of mother figure. Peter takes the Darlings on several adventures, the first truly dangerous one occurring at Mermaids' Lagoon. At Mermaids' Lagoon, Peter and the Lost Boys save the princess Tiger Lily and become involved in a battle with the pirates, including the evil Captain Hook. Peter is wounded when Hook claws him. He believes he will die, stranded on a rock when the tide is rising, but he views death as "an awfully big adventure". Luckily, a bird allows him to use her nest as a boat, and Peter sails home.
In gratitude for saving Tiger Lily, her tribe guard his home from the next imminent pirate attack. Meanwhile, Wendy begins to fall in love with Peter, at least as a child, and asks Peter what kind of feelings he has for her. Peter says that he is like her faithful son. One day while telling stories to the Lost Boys and her brothers, John and Michael, Wendy recalls her parents and then decides to take them back and return to England. Unfortunately, and unbeknownst to Peter, Wendy and the boys are captured by Captain Hook, who also tries to poison Peter's medicine while the boy is asleep. When Peter awakes, he learns from the fairy Tinker Bell that Wendy has been kidnapped â in an effort to please Wendy, he goes to drink his medicine. Tink does not have time to warn him of the poison, and instead drinks it herself, causing her near death. Tink tells him she could be saved if children believed in fairies. In one of the play's most famous moments, Peter turns to the audience watching the play and begs those who believe in fairies to clap their hands. At this there is usually an explosion of handclapping from the audience, and Tinker Bell is saved.
Peter heads to the ship. On the way, he encounters the ticking crocodile; Peter decides to copy the tick, so any animals will recognise it and leave him unharmed. He does not realise that he is still ticking as he boards the ship, where Hook cowers, mistaking him for the crocodile. While the pirates are searching for the croc, Peter sneaks into the cabin to steal the keys and frees the Lost Boys. When the pirates investigate a noise in the cabin, Peter defeats them. When he finally reveals himself, he and Hook fall to the climactic battle, which Peter easily wins. He kicks Hook into the jaws of the waiting crocodile, and Hook dies with the satisfaction that Peter had literally kicked him off the ship, which Hook considers "bad form". Then Peter takes control of the ship, and sails the seas back to London.
In the end, Wendy decides that her place is at home, much to the joy of her heartsick mother. Wendy then brings all the boys but Peter back to London. Before Wendy and her brothers arrive at their house, Peter flies ahead, to try and bar the window so Wendy will think her mother has forgotten her. But when he learns of Mrs Darling's distress, he bitterly leaves the window open and flies away. Peter returns briefly, and he meets Mrs. Darling, who has agreed to adopt the Lost Boys. She offers to adopt Peter as well, but Peter refuses, afraid they will "catch him and make him a man." It is hinted that Mary Darling knew Peter when she was a girl, because she is left slightly changed when Peter leaves.
Peter promises to return for Wendy every spring. The end of the play finds Wendy looking out through the window and saying into space, "You won't forget to come for me, Peter? Please, please don't forget."

Question: What does Wendy beg of Peter?
Answer: to not forget to come for her every Spring
Alternative answer: Not to forget to come to her.