This question is quite simple but at the same time quite interesting, morally. For the longest time, Americans weren't "allowed" to travel to Cuba, and yet would a dual citizen be considered guilty of a crime for traveling there on his/her other passport? What about an American/Cuban dual citizen? Does the United States restrict where its citizens are allowed to travel?
The restrictions on travel to Cuba have always been rooted in sanctions in the form of a trade embargo. The US government has long restricted its citizens from spending money with certain individuals or entities abroad (see, for instance, the Trading with the Enemy Act of 1917). This impacts travelers since virtually every trip will involve spending at least some money (there's an airport tax for one thing). You can see our previous question Travel to Cuba as dual citizen (USA+other)? for information on that situation.
Starting last year, there are also travel restrictions to North Korea, which apply to US passports:
Travel to, in, or through North Korea on a U.S. passport without this special validation may justify revocation of your passport for misuse under 22 C.F.R. § 51.62(a)(2) and may subject you to felony prosecution under 18 U.S.C. § 1544 or other applicable laws.
There's a procedure to apply for a Special Validation Passport if you think you have an acceptable reason. There are also financial sanctions that apply to North Korea.
The regulations for passports permit the Secretary of State to issue other restrictions on where US passports may be used in the case of war, armed hostilities, or imminent danger to health or safety.
The US government issues travel advisories for their citizens via the State Department, and they are listed at this website.
Each country is ranked from 1 to 4, 4 being the most restrictive suggestion of "Do Not Travel".
As of this writing, there are 11 countries rated 4:
- Central African Republic (CAR)
- South Sudan
- North Korea
US citizens have never been banned from visiting Cuba. Commercial embargoes meant that it was difficult to get flights to Cuba, but this did not prevent travel there.
As there are no exit (immigration) controls at US borders, effectively the US cannot ban you from leaving. It is upto the receiving country to stop you from entering.
However the State Department (well, the Secretary of State) has the right to issue restrictions on US passports in times of war, armed conflict, or if there is imminent danger to health and safety.
In practice, this usually means you require special permissions, such as the case of North Korea:
Individuals cannot use a U.S. passport to travel to, in, or through North Korea without a special validation from the Department of State.