...is generally presumed to be whoever was writing that part of the Red Book (that being "The Downfall of the Lord of the Rings and the Return of the King" along with appended translations and notations).  Which generally meant it was either Bilbo (in the case of the Hobbit and certain translations), Frodo (for LotR), or Sam (for the parts that came after).  Tolkien liked to treat the whole thing as if it were an historical document that he (or someone in the past) had found and translated.  Sometimes, he is very explicit about the identity of the narrator (in Unfinished Tales, in the chapter "The Quest of Erebor," he makes it very clear that though the person actually telling the tale is Gandalf, Frodo is the narrator recording it); more often than not, he isn't.