Can I claim Irish citizenship through a great-grandparent?

Not directly. A great-grandparent born in Ireland is not enough on its own. The claim works only if the generations between you behaved in time: your parent must have been entered on the Foreign Births Register before you were born, claiming through their own grandparent. If they were, they were an Irish citizen at your birth, and you can register in your turn. If they never registered, or registered after you were born, the chain is broken and cannot be repaired backwards. This is where most great-grandparent claims end. It is also why families who may want the option register each generation promptly, even when no passport is needed yet.

Useful pages: Irish in USA · The Irish surnames map · Trace your Irish roots · Coming home · Irish around the world

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