A naturalized American citizen (or country of the European Union) has some limitation to obtain visas to other countries?
closed as too broad by drat, JoErNanO♦, CGCampbell, Willeke♦, Gagravarr Dec 9 '15 at 9:24
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Normally it does not matter. However, it can. Try getting a visa to a bunch of Arab nations with a place of birth of "Jerusalem"! (Even if you were the child of a diplomat stationed there.)
Various visa application forms require the traveler to list all citizenships, as well as things like previous visa rejections or criminal convictions. So unless the traveler is lying -- a very bad idea -- the consular officials will know.
And I believe that both the US and EU apply special scrutiny to travelers born in some countries perceived as high risk, regardless of their current citizenship. (Whether that is justified or not belongs on the politics SE site.)
Follow-Up: Check the recently proposed changes to the US visa waiver program: BBC - US House votes to restrict visa-free travel after Paris attacks
I was just looking at some visa applications that asked for nationality-at-birth. And in the case of people who hold dual citizenship, whether they know or want it or not, they may not even need a visa to visit a country which thinks they are still a citizen.