Even if a child was not born in the US., he or she may still be US citizen, if one or both parents were citizens at the birth of the child.
There are different sets of rules for parents passing on American citizenship to their children born outside the US, depending on the child's date of birth. For children born on or after November 15, 1986, the rules are as follows - If both the child's parents are US citizens at the the time of birth, the child automatically gains citizenship with no strings attached. However, if only one parent is a US citizen, this parent can pass on citizenship only if he or she resided in the US for at least five years. Evidence of residency may include tax returns, bank statements, apartment lease, employment records, and letter from an employer stating length of employment etc.
Working on the assumption that the parent meets the residency requirement, the parent files a Report of Birth Abroad of a Citizen of the United States of America at the US consulate in the country of the child's birth. This will confer American citizenship on the child, and will allow for a US passport to be issued in his or her name.
The child will also receive a certificate of consular registration of birth, which will serve as conclusive evidence of his or her US citizenship in future years.
Upon returning to the US., the parent can apply to the USCIS for a certificate of citizenship for the child using Form N-600.