
Samsung Galaxy Note3
The latest firmware for Samsung Galaxy Note3 (SM-N900U) is N900UZTUBPG2 with Android 5.0 (Lollipop), released on July 12, 2016 for 1 region (CSC). Every official firmware build for SM-N900U, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/N900UZTUBPG2
SM-N900U started on Android 4 and has received 1 major Android upgrade; it now runs Android 5.0 (Lollipop). The newest build, N900UZTUBPG2, was released on July 12, 2016. No new build has appeared for 125 months, so regular updates for this model have most likely ended.
Specifications of SM-N900, a regional variant of the same device. Chipset and SIM lines are left out: they can differ by region.

SM-N920AZW

SM-N920VZW

SM-N950UZK

SM-N920AZK

SM-N950X

SM-N930V

SM-N9300

SM-N930S
The newest firmware for SM-N900U runs Android 5.0 (Lollipop): build N900UZTUBPG2, released on July 12, 2016.
The latest build for SM-N900U, N900UZTUBPG2, was released on July 12, 2016 (10 years ago).
SM-N900U has had no new firmware for 125 months, so regular updates have most likely ended.
Yes, all firmware files for SM-N900U are 100% official, original, and obtained directly from Samsung servers.
To flash SM-N900U, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing SM-N900U.