On July 1, 2001, a new Citizenship Act came into effect in Sweden. The new law made it possible to become a citizen of another country without losing your Swedish citizenship. The law was further amended on April 1, 2015.
As of July 1, 2001, a Swedish citizen who acquires citizenship in another country will be allowed to keep the Swedish citizenship, if the other country permits it. By the same token, if you become a Swedish citizen you can keep your foreign citizenship if the laws of that country permit it.
Those who lost their Swedish citizenship as a result of acquiring citizenship in another country before July 1, 2001, were able to regain their Swedish citizenship through notification before July 1, 2003.
On April 1, 2015, the law was further amended and now anyone who lost his/her Swedish citizenship due to applying for another country's citizenship may submit a request to become a Swedish citizen again.
More information is available on Regaining Swedish citizenship.
U.S. immigration law states that all U.S. citizens, including those who have dual citizenship, must be in possession of a valid U.S. passport when leaving and entering the U.S. It is thus advisable for dual citizens to bring both passports when traveling between the two countries.