Motorola has posted release notes for the Android 5.1 update for both the first-gen Moto X, as well as the second-gen AT&T and Pure Edition variants.
While it doesn't appear that the updates are rolling out to Moto Xs just yet, this is a significant step forward for the above devices. You can check out the release notes in full from the source link below, but they cover bits of Android 5.1 Lollipop that should be mostly familiar by now. While Motorola doesn't list a build number for the first-gen Moto X, you'll be looking out for software versions 23.11.21 and 23.11.14 for the AT&T and Pure Edition second-gen models, respectively (if you're interested in such things).
In any event, the appearance of the release notes should indicate that the updates will be coming fairly soon. If you happen to snag the update, be sure to drop by our forums, or shout out in the comments below how the update is treating you.