In the Forgotten Realms, there are 5 factions:
These factions are outlined here: https://www.dndbeyond.com/sources/dnd/basic-rules-2014/appendix-c-the-five-factions
According to ALPG 15.3, all characters can join one of the above 5 factions.
Factions used to play a larger role in older adventures (for example, the DDEX03 and DDAL05 series). But in recent years, the AL admins have removed any mechanical benefits from being in a faction.
As a faction member, the following is valid:
There are no out-of-adventure mechanical benefits to being part of a faction and gaining renown. For example, there’s no extra training or magic items you can gain.
It’s mostly for fun and role playing purposes.
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