I was born in the US, and at the time my father was a Mexican immigrant serving in the US Army who married my mother, a US citizen, just prior to my birth. Thus I had dual US/Mexican citizenship until I turned 18 years old. At that time, under US law, in order to keep my US citizenship I was required to relinquish all other foreign citizenships.
While some countries allow adults to maintain multiple citizenships, the US does not. When you become a US citizen, you are required to reject allegiance to all foreign governments.