In Norse mythology, Váli is a son of the god Odin and the giantess Rindr. Váli has numerous brothers including Thor, Baldr and Víðarr. He was birthed for the sole purpose of avenging Baldr, and does this by killing Höðr who was an unwilling participant, and binding Loki with the entrails of his son Narfi. He grew to full adulthood within one day of his birth,and slew Höðr, before going on to bind Loki. Váli is prophesied to survive Ragnarök.
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Often incorrectly referred to as the son of Loki, though this is most likely an early transcription error. This misconception is based on a single passage containing the phrase "Then were taken Loki's sons, Váli and Nari" in Gylfaginning, which also describes Váli as the son of Odin in two instances All other historical documents found at this time ascribe Váli only the role of Odin's son, with the exception of transcripts based on the original misattribution.
Myths
The Váli myth is referred to in Baldrs draumar:
Rindr will bear Váli in western halls; that son of Óðinn will kill when one night old— he will not wash hand, nor comb head, before he bears to the pyre Baldr's adversary. - Ursula Dronke's translationIn Völuspá:
There formed from that stem, which was slender-seeming, a shaft of anguish, perilous: Hǫðr started shooting. A brother of Baldr was born quickly: he started—Óðinn's son— slaying, at one night old.And in Prose Edda Gylfaginning (where he is ascribed the role of Loki's son)
Now Loki was taken truceless, and was brought with them into a certain cave. Thereupon they took three flat stones, and set them on edge and drilled a hole in each stone. Then were taken Loki's sons, Váli and Nari or Narfi; the Æsir changed Váli into the form of a wolf, and he tore asunder Narfi his brother. And the Æsir took his entrails and bound Loki with them over the three stones: one stands under his shoulders, the second under his loins, the third under his houghs; and those bonds were turned to iron. — Arthur Gilchrist Brodeur's translationThe Prose Edda also mentions him again. Gylfaginning contains this passage:
"One is called Ali or Váli, son of Odin and Rindr: he is daring in fights, and a most fortunate marksman."
The same text also states that he will survive Ragnarök, along with his brother Víðarr and the sons of Thor, Móði and Magni.
Parentage
Due to early mistranslation or confusion leading to the single description of Váli as the son of Loki "Þá váru teknir synir Loka, Váli ok Nari eða Narfi." from the Prose Edda translated as "Then were taken Loki's sons, Váli and Nari" we receive the only reference to Váli as the son of Loki, while even the same text refers to Baldr's death being avenged by his brother (in Völuspá S33) as well as Váli being the Son of Odin in S51, something repeated in Baldrs draumar.
In the late period Gesta Danorum we also see that Odin is said to have a son with Rinda that will avenge his other son, Baldr's, death - though in this case the name of this new son is Bous rather than Váli. In all these tales Odin goes out immediately - either through seduction, deception, or force - to sire this son (Something that would not have been needed if his goal had been only the destruction of the blind and defenseless Hoth.)
Similarly where each of these documents ascribe Váli the role of Loki's son we see only in the postscript or translation notes that this transformation was a punishment when in fact the gift of wolf's strength and rage is well attested as being granted by Odin to warriors known as Ulfhednar, which would make his son Váli a Berserker and a possible origin for the ulfhednar legend.
Finally we see a different description in Hauksbók in this version of "Völuspá" stanza 34 begins: "Þá kná Vála | vígbǫnd snúa", usually amended to the nominative Váli in order to provide a subject for the verb; in Ursula Dronke's translation in her edition of the poem, "Then did Váli | slaughter bonds twist". This presumably refers to Váli, son of Óðinn, who was begotten to avenge Baldr's death, and thus not unlikely that he bound Loki; while it is highly improbable that it refers to Váli, son of Loki, who is attested nowhere but one line of the Prose Edda - which itself confirms the existence of Váli as son of Odin and avenger of Baldr in two locations