The movie is a fictionalized version of the legend of Anastasia, a daughter of the Russian royal family who allegedly escaped while the rest of her family was killed.
The movie starts in the past, with the royal family receiving a curse from the evil Rasputin, which leads to the uprising of the nation and the murder of the whole royal family except for the grandmother and Anastasia, who are saved by a kitchen boy. They try to board a train to escape, but Anastasia falls off and is left behind.
Fast forward ten years later, and we meet an orphan called Anya who has a necklace exactly like Anastasia and who doesn't remember who she is. She wants to go to Paris to find her family, and so joins up with two con artists, Dimitri and Vlad (?), who realise that she really resembles the princess and would be a perfect candidate to fool the grandmother into giving them the reward for finding Anastasia. They pretend that they really think that she is Anastasia to get her to come.
They teach her everything about Anastasia and go to Paris to meet with the grandmother. Dimitri discovers that Anastasia is actually the princess because she remembers the kitchen boy (actually, him) when he hadn't told her. He also realises that he loves Anya, but resolves to give her back to her family.
Anya realises that Dimitri and Vlad were only in it for the money, and tries to leave. Dimitri forces her to meet with the grandmother, and so the family is reunited, while Dimitri refuses any reward, and disappears
Rasputin reappears and tries to kill Anya, who is saved by Dimitri, and they both run off together and live happily ever after...