Generally, immediate relatives of US citizens do not have very many barriers in the way of their attaining US citizenship and children of citizens are no different.
Section 320 of the Immigration and Nationality Act, amended by the Child Citizenship Act, puts forth requirements for children to qualify for US citizenship. In order to do so, a child must have at least one parent who is a US citizen by naturalization or birth, be under 18 years old and be living in America in the custody of the citizen parent.
Those who fulfill these requirements before they turn 18 will automatically receive citizenship without having to file an application. Still, they will have to file Form N-600, Application for Certificate of Citizenship to receive their documentation.
There are a few things that people should keep in mind when considering this path to citizenship, however. Notably, in order to be considered a child, a person is not allowed to be married. In addition, those who were born out of wedlock need to be legitimated before they turn 16 years old.
Adopted children can also attain US citizenship through a parent but they may have to meet additional requirements.