James' happy life at the English seaside is rudely ended when his parents are killed by a rhinoceros and he goes to live with his two horrid aunts. Daringly saving the life of a spider he comes into possession of magic boiled crocodile tongues, after which an enormous peach starts to grow in the garden. Venturing inside, he meets not only the spider but a number of new friends including a ladybug and a centipede who help him with his plan to try to get to New York.

The movie is about a boy with a unique aging disorder: one that makes him age 4 times faster than normal. It picks up when Jack (Robin Williams) is 10 years old, but looks 40. He tries to go to public school for the first time, and to become friends with kids his own age. His physical appearance causes him lots of problems, however.