The United States — and other countries, such as Australia, Canada, and the United Kingdom — permits dual citizenship. Some countries, such as India and Japan, however, do not allow dual citizenship. In fact, they require their citizens to relinquish their citizenship in those countries once they become Americans.
U.S. citizens who are also citizens of other countries will be required to use their U.S. passports when traveling in and out of the United States.
It’s best to check the dual citizenship policy of a country (if intending to maintain citizenship in that country) before applying for U.S. naturalization.
Here’s some additional information from the U.S. Department of State.