It’s been a very long road, fraught with misunderstandings, drama, and even gloating by the competition– but we are now at an end for it. Bethesda have announced on Twitter that the next update for Fallout 4, due later this month, will enable mods for the PS4 version of the game. At last.
These mods will be far more limited than they are on Xbox One and, obviously, PC- they will have a total file size limit of 1GB, and they will only use assets available within the game files. This was a compromise Bethesda reached with Sony after Sony initially apparently blocked the release of mods on the system at all.
As of right now, only the Fallout 4 mods have a concrete release date on PS4- for Skyrim remastered, the mods will presumably be enabled on the PS4 at a later date. Fallout 4 and The Elder Scrolls Skyrim: Special Edition are both available on Xbox One, PS4, and PC.