updated 10:40 am EDT, Fri March 21, 2014
Current technique limited by processor
A hacker going by the alias "winocm" has demonstrated what appears to be the first untethered jailbreak of iOS 7.1. The hack currently works only with the iPhone 4 however, since it's the only iOS 7-capable device with an A4 processor. winocm and well-known jailbreaker iH8sn0w are said to be working on a jailbreak for A5-based devices, like the iPhone 4S, iPad 2, and first-generation iPad mini. That could mean it will be some time before modern A7-based devices are cracked.
"Untethered" implies a device doesn't have to be connected to a Mac or PC whenever it's restarted. Although it may be a while before A7 support arrives, iOS 7.1 was released just last week, making the progress of the jailbreak so far relatively rapid.