The United States taxes anyone who qualifies for having to file a form 1040 (as opposed to 1040NR) on their worldwide income. It does not matter if the person resides in the United States and earns income abroad, or if the person resides abroad and earned income from countries that are not the United States. You still have to report all of the income on your U.S. Tax Return when you qualify as a U.S. person.
It should be noted that you may receive a Foreign Tax Credit for taxes paid in a Foreign Country, as well as you may qualify for the Foreign Earned Income Exclusion if you meet either the Bona-Fide Resident or Physical Presence Test. Sharma Marcus can help you navigate the complexities involved in the Global Reporting of your Income.