How do you pronounce Nina Ananiashvili's last name? And how would you address someone of such fame if you met her/him? By full name or along the lines of Mr. and Ms./Mrs./Miss?

The Russian speakers on this board will correct me if I'm wrong, but I believe it's pronounced ah-nan-YASH-vee-lee. If you were speaking to her in Russian, you'd probably say Nina Father'sName-ovna, but over here it's perfectly fine to say Miss/Ms. Ananiashvili.

Is there a chance that you'll be meeting her soon, Old Fashioned?

here's my try:
NEE-nah ah-nahn-YASH-vee-lee
i think russian dancers are used to hearing americans address them as is usually done in English and in the States, therefore: Miss Ananiashvili should be accepted as respectful and polite.
i am told that addressing such a russian by her first and her patronymic names is most polite of all. for example, i was told, it was much more polite to address, say, Galina Ulanova as Galina Sergeyevna than to address her as Mme (or Miss) Ulanova. still i think Ulanova would have realized that the latter form of address was meant with all due respect and politeness.
that said, i do not, tho' i suspect some who read this site do, know Nina Ananiashvili's patronymic. so i can't help with that in her case.

You always refer to a ballerina as a miss, even after several weddings.....

Thanks everyone. I hope I do get to meet her tomorrow, but it's not certain.