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Here's a list of online genealogy resources on the archives.gov website: Online Databases - use for free at the National Archives
And here's a place to start on FamilySearch: Ukraine Online Genealogy Records Genealogy - FamilySearch Wiki
I hope that helps!
Kelly, Thanks for your help the FamilySearch Wiki is particularly helpful.
That was the best help in locating Genealogy records for Ukraine I have seen yet. Thank you David for asking that particular question, and Thank you Kelly for the great answer.