Got the American nationality also obtained the passport as I was born there only but I have never visited USA, and I do not have social security number.
My question is: With the new law (FATCA), do I have to pay taxes or not?
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Yes, you do. Every American citizen does. The new law has nothing to do with it, you were subject to the American taxation your whole life. The new law just makes it harder for you to ignore it.
You'll have to file a yearly tax return (IRS form 1040), and check, considering the foreign income exclusion (form 2555) and foreign tax credit (form 1116) whether you actually owe any money to the US government or not. If you do - you have to pay.
Don't forget the FBAR and other various forms that exist to punish you for being a US citizen.
Read more in the IRS Publication 54.
Tax obligations travel with American nationality. If you don't live in the United States, the first $90k or so of your income is not taxed, but you still have to file a return. Sounds like you acquired a rather expensive passport.
ps FATCA has nothing to do with your obligation to file a tax return. You do need to report foreign bank accounts if you have $10k or more in the account at any time during the year.