AT&T and Verizon, with their insistence on locked bootloaders for Android devices, are the scourge of the Android customization scene. Unfortunately they're also the largest carriers in the United States, which leaves a lot of Android power users in a pickle. If you're on either carrier and rocking a branded Galaxy S5, today is your lucky day: someone's gone and made a near-universal and amazingly simple root method that should work for the S5 (and more) on both carriers.
"Towelroot" comes from noted XDA developer Geohot, and according to him, the simple one-click APK root will absolutely work on the AT&T and Verizon Galaxy S5, the AT&T Galaxy S4 Active, and the Nexus 5. Just install the downloadable app, tap "make it ra1n," and wait. Support for the AT&T and Verizon variants of the Galaxy Note 3 has not been verified, but might be possible. [Editor's note: we tried it on a stock AT&T Note 3 with no success.] In fact, Geohot claims that any Android device with a kernel built before June 3rd (anything running 4.4.2 or lower, basically) should theoretically be rooted by Towelroot.
Definitely invited: AT&T GS5, Verizon GS5, GS4 Active, Nexus 5
May have some troubles at the door but invited: AT&T/Verizon Note 3
Possibly invited: Every Android phone with a kernel build date < Jun 3
Geohot notes that applying this root to a Samsung device with the Knox security system in place will almost certainly void your warranty, but if you want root that bad, you probably don't care. Anyone having issues can contact the creator directly at email@example.com, and he is gracefully accepting PayPal donations at the Towelroot website.
Update: AT&T Note 3 with kernel version 3.4.0-722276 is confirmed as incompatible with Towelroot, at least in the current release.
Update again: It looks like he worked out the kinks. Yup, it works with the AT&T Note 3 now.
Killed bugs: Note 3 with kernel 3.4.0-722276 is FIXED! Redownload. Known bugs: su binaries don't seem to update. This may not be the root's fault. Moto and HTC don't currently work because /system is write protected.