technically, baudehlo is mostly correct. Most new phones: androids (HTC, Samsung, etc), iphones and whatever, are able to be used around the world as long as they're unlocked and have SIM card slots because these countries are at least using one or more of these frequencies: 850/900/1800/1900/2100, which most new phones are capable of.
The parts that he's wrong would be those few "dumb" phones that companies still sell that aren't able to use the other frequencies, but this is rare. Also, some countries don't use the same "LTE" or "4g" frequencies that the US uses, so if you were to bring your phone there, it would still work, but it be limited to "3g" instead of "4g" speeds, but this varies country to country and it also depends on the network and plan that you choose.
The parts that he's wrong would be those few "dumb" phones that companies still sell that aren't able to use the other frequencies, but this is rare. Also, some countries don't use the same "LTE" or "4g" frequencies that the US uses, so if you were to bring your phone there, it would still work, but it be limited to "3g" instead of "4g" speeds, but this varies country to country and it also depends on the network and plan that you choose.