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Well, this is of course based on a popular legend which is itself based on real historical events in Russian history. The legend has been retold in various ways many times, in different movies and whatnot, sometimes sticking relatively close to the story's roots, and sometimes not. (I can't really think of the story of Anastasia without thinking of the movie "Candleshoe," for example.) In fact, the basic premise surely goes back much farther than the story of Anastasia particularly; I mean, how common is it, particularly in fairy tales, for some peasant to turn out to actually be long lost royalty? But anyway, I'm getting ahead of myself.
The movie is narrated by Dowager Empress Marie Feodorovna (Angela Lansbury). It starts in 1916, when her son, Tsar Nicholas II, was hosting a ball in honor of the 300th anniversary of the Romanov dynasty (which actually I think would've been three years earlier). There's a brief, sweet little scene during the ball, between Marie and her eight-year-old granddaughter, Anastasia. Then suddenly, Grigori Rasputin (Christopher Lloyd) shows up. He was a monk/mystic who had apparently been an ally of the Romanovs, until having been banished as a power-mad traitor. But Rasputin sold his soul to gain the power to put a curse on the Romanovs, to entirely eliminate the family. (I'd really like to learn more about actual Russian history someday; I mean more than the tidbits I glean from skimming Wikipedia to aid me in writing this review. I'm pretty sure the timing of events in this movie is a bit off, things condensed for the purposes of convenient storytelling. I'm also pretty sure that Rasputin didn't have any actual magical powers. And even if he influenced the Russian people against the Romanovs, I suspect the tsar and his family may not have been blameless in their own downfall. Oh, and I rather doubt that the real Rasputin had a funny talking bat named Bartok as a minion.) In any event, at some point (perhaps sometime the next year, as the real Russian Revolution was in 1917) there was a revolt against the Romanovs, and most of them were killed (this didn't happen until the year after the revolution; like I said, it's all condensed). Marie and Anastasia escaped (they were aided by a kitchen boy), though the movie shows Rasputin pursuing them before falling through a patch of ice to drown. (Which I'm quite sure is not how he died in real life.) But then Anastasia was separated from her grandmother.
The movie then flashes forward ten years. St. Petersburg is abuzz (in a lavish song and dance number) with the rumor that Anastasia is alive. A young man named Dimitri (John Cusack) and his friend Vladimir (Kelsey Grammer) have a plan to find some girl and teach her to pretend to be the Grand Duchess Anastasia, and take her to Paris, where Marie is offering a huge reward to anyone who can reunite her with her granddaughter. Meanwhile, we see a young woman named Anya (Meg Ryan), an orphan who doesn't remember who she was before being taken in by... I dunno, some nasty old woman, ten years earlier. She does, however, have a necklace that says "Together in Paris," which... was the key to a music box which we had seen Marie give Anastasia earlier (and of which Dimitri is now in possession). Anyway, one day she's trying to decide whether to take an unappealing job or to seek out whoever had given her the necklace (wanting desperately to find out her own identity and whether she has a family). While she's waiting for a sign, a puppy named Pooka urges her towards St. Petersburg (by tugging her scarf; it's not a talking dog, thankfully).
Well, Anya finds out that she needs travel papers to go to Paris, and someone tells her to find Dimitri to help with that. Meanwhile, Dimitri and Vlad have pretty much given up on auditioning girls to play Anastasia, because none of them are anything like Anastasia. They've been holding these auditions in the remains of the old palace, and when Anya shows up looking for Dimitri, she starts remembering the place, but still not who she was. Then when she finally meets Dimitri, he tells her they're planning on going to Paris with Anastasia. He also convinces her there's a possibility she might actually be Anastasia, since she looks so much like her. Though he doesn't tell her about his and Vlad's scam. So, the three of them go to Paris together. Meanwhile, Bartok (Hank Azaria) witnesses the whole thing, and then this vial of goo that was the source of Rasputin's power comes to life, and drags Bartok down to some underworld limbo, where he's reunited with his old master. When Bartok tells Rasputin that Anastasia is alive (which must be the case because the goo got all glowy and active when Anya was nearby), Rasputin is eager to kill Anastasia, to complete his unfinished curse. (Though it's odd that he seems to have no interest in killing Marie. Maybe because she wasn't a Romanov by blood, I guess.)
Dimitri and Vlad spend the trip training Anya, and it's fairly clear that in spite of initially not getting along, they gradually fall in love, though neither one says so. (Vlad, of course, realizes it immediately.) When they finally reach Paris, they go to see a woman named Sophie (Bernadette Peters), who is Marie's first cousin and lady-in-waiting. Sophie and Vlad are old friends (maybe more), so Sophie agrees to help, even though Marie had just recently said she was tired of seeing so many impostors pretending to be Anastasia, and won't see anyone else claiming to be her granddaughter. Still, something Anya said during Sophie's interview made Dimitri finally realize the truth....
Ah, well, I've said too much already. I won't give away any more details. But of course there are misunderstandings and obstacles to be overcome, including some rather cliched stuff, as well as Rasputin's continued attempts to kill Anastasia (and there's some rather creepy imagery involved, which might be a bit scary for young children). And of course there's ultimately a happy ending (or beginning, depending on how you look at it). Anyway, I definitely thought it was a good movie. The songs all worked really well in their respective scenes. There was some decent humor (most of it from Bartok; it's not so much what he said as how he said it). And most importantly, the story was pretty good. Plus, I thought Anya was awfully cute and likable, in a scrappy-but-sweet orphan kinda way.