evasi0n7 can jailbreak iOS 7.0-7.0.6 untethered. p0sixspwn can jailbreak iOS 6.1.3-6.1.5 untethered (see here for 6.1.6).
How to jailbreak iOS 7.0-7.0.6 (and fix problems). iOS 7.1.x is not jailbreakable, except an unsupported tethered iPhone 4 jailbreak.
Restoring or updating will update your device to iOS 7.1.1 and remove your jailbreak. There is no way to downgrade from iOS 7.1.x to 7.0.x.