The Moto X (2014) and the Moto X (2013) handsets are currently receiving the Android 5.1 Lollipop update. In the recent past, Android 5.1 soak tests were rolled out for both models through U.S. carriers. The Android 5.1 Lollipop is expected to improve the overall performance of the Moto X (first generation) and (second generation) by fixing the issues that arrived with Android 5.0 Lollipop OS.
The Motorola product page has the up-to-date information on the version of Android Lollipop available for different Motorola Moto devices such as Moto X (2013), Moto X (2014), Moto G (2013), Moto G (2014), Moto E and so on across different U.S. carriers.
Moto X (1st Gen)
According to Android Pit, the Android 5.1 Lollipop update soak test has begun for AT&T and Verizon users of Moto X (2013). Rolling out of the soak test indicates that the actual 5.1 Lollipop will be made available for AT&T and Verizon users very soon. As of this writing, there is no confirmation on when the T-Mobile and Sprint variants of Moto X (Gen 1) users will receive the update.
Moto X (2nd Gen)
David Schuster, Motorola’s senior director of software product management, has declared on the Google+ page that the Android 5.1 Lollipop update has reached full deployment stage for AT&T and Verizon users of Moto X (2014).
Another Android Pit report claims that the users of Moto X (second generation) Pure Edition available through T-Mobile have started receiving the Android 5.1 Lollipop update from June 26. U.S. Cellular has already rolled out the Android 5.1 Lollipop OTA update for its users with build number LPE23.32-21. Users can check for the update by navigating to Settings > About Phone > System Update.
In order to update the latest Android firmware on Moto X (first generation) and Moto X (second generation) devices, users are first required to install the most recent version of Motorola Update Service which is available at Google Play Store. This service automatically notifies Motorola smartphone users on the arrival of new firmware updates.