In Elijah's case, we have the recorded instance of a man being translated to heaven apart from a "normal" death (the only other such case was Enoch, although the Scriptures don't give details on how he was taken). We also have Malachi's prophecy (3:5-6) concerning the return of Elijah to prepare the way of Messiah by restoring the hearts of the fathers to sons, etc. Some connected this promise with "My messenger" spoken of in Malachi 3:1ff. On this basis, when John the Baptist began his public ministry people came out to inquire, "Are you Elijah?"

Interestingly, Jesus said that John the Baptist "was Elijah, if you will receive it." By this I take it that He meant John fulfilled the Messenger portion of the prophecy (preparing the way for Messiah), in the spirit and power of Elijah - not that John was Elijah reincarnated. Of course, Elijah also showed up at the Mt. of Transfiguration, but that was not a "public viewing" as only Yeshua and His three closest disciples saw him.

You will recall the other questions asked of John were, "Are you the prophet?" (referring to the one "like unto" Moses prophesied in Deut. 18:18; Jesus is said to be "counted worthy of more glory than Moses" in Hebrews 3:2-5), and "Are you the coming one?" (i.e., Messiah).

Though no one was expecting Moses to show up again, he did - at the Mt. of Transfiguration (which makes, "Did Moses ever set foot in Canaan?" a nasty Bible trivia question). It also raises the question of what "body" did he inhabit, since he seemingly died a natural death (though no one knows the place of his burial), and does not yet have a resurrection body (according to 1 Cor. and 1 Thess., we get those at the Lord's return).

Though the word "resurrection" is not found in the Hebrew Scriptures (Old Testament), the idea of life after death is hinted at as early as Job (who likely was a contemporary of Abraham) when he said, "Even after my skin is destroyed, yet from my flesh I shall see God" (Job 19:26). Daniel 12:2 makes specific reference to the dead returning to life: "And many of those who sleep in the dust of the ground will awake, these to everlasting life, but the others to disgrace and everlasting contempt." [We also have the example of a corpse being resuscitated after contacting Elisha's bones (2 Kings 13:21).] Still, liberal Judaism today - like the Saduccees of Jesus' day - deny the possibility of life beyond the grave.

Interestingly, when Yeshua asked His disciples what they were hearing concerning His identity, they responded, "Some say John the Baptist (who had already been beheaded); others say Elijah; and still others, that one of the prophets of long ago has come back to life" (Luke 9:18). So resurrection seems to have been viewed as a real possibility, not just theory, among those who followed the Pharisees' teaching.

The idea of reincarnation (gilgul nephesh, or transmigration of souls) is a later development in Judaism (perhaps by the Karaites; see
http://www.amfi.org/mailbag/reincarnation.htm). The idea persists in some branches of Hasidism, but has no biblical basis.