I have an iPhone but no longer under a contract?

I have an iPhone 5 and am no longer under a contract. Can I sign up for another carrier but still keep the same phone? I'm thinking about signing up for a different carrier and don't want to buy another iPhone. Thanks!

Added (1). Also which carriers would let me do this?

You may need to pay a small fee to switch it over

Perhaps. It can work if both carriers use the same technology (CDMA or GSM). Ask the carrier you wish to switch to.