Is it possible to take an iPhone bought in the US, replace the SIM card with a European one, and have it work in Europe?

Unfortunately I think they’re SIM-locked and the SIM has to be from the carrier the phone is locked to.

I’m pretty sure the iPhone is locked to a particular carrier, isn’t it? With most phones, as long as it’s 3 or 4 band and has an unlocked SIM, you can travel anywhere; however, the iPhone is funky… so I’d ask your carrier first. If anything, you could buy a SIM there (Vodafone are great) and a cheap pay as you go phone. In Greece places like Germanos have them for 10–20 euro or so.

The phone is locked to At&t, you have to get it unlocked to use it on a network overseas, however this is not in At&t’s best interest because they’re losing money in the process, so they will not unlock it for you. You have to go to a 3rd party wireless provider to get it unlocked, or google how to unlock the phone yourself. If you do unlock the phone it will void the one year manufacturers warranty provided by Apple.