I am a US citizen living outside of the US. Neither my wife nor my children is a US citizen, nor do they have SSNs.
Am I allowed to list them as dependents?
May I file as head-of-household, listing my children as dependents for this purpose?
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In order for you to claim someone as your dependent, they must be a US citizen, US national, US resident alien, or a resident of Canada or Mexico. See Form 1040 instructions, section "Who Qualifies as Your Dependent?", subsection "Is Your Qualifying Child Your Dependent?". (And for non-child dependents, the same requirement can be found in "Is Your Qualifying Relative Your Dependent?".)
If your children are really not US citizens, and they do not have green cards and have not spent significant time in the US, then they are not resident aliens either. So unless they are residents of Canada or Mexico, you cannot claim them as dependents.
There is one exception which is if your children are adopted. In the section "Exception to citizen test", it says that if you are a US citizen or national and your children lived with you all year as a member of your household, they qualify as dependents on the same basis as US citizens.