The shards became an heirloom to the heirs of Elendil and were kept in Rivendell by Elrond for safekeeping. They were eventually reforged during the Third Age by the Elves of Rivendell.
The reforged sword was called Andúril 'Flame of the West' and was carried by Aragorn.
How Elendil came to be the bearer of Narsil is unknown, but we do know that the sword was first created by Telchar of Nogrod, who also made the dagger Angrist.
"In this elvish sheath dwells the Blade that was Broken and has been made again. Telchar first wrought it in the deeps of time . . ." [TTT II, ch. 6, Aragorn].
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