I've heard that the second Vatican council eliminated the concept of limbo. Or maybe that it never existed to begin with.
Was limbo part of "official" Catholicism before Vatican II? What changes were made at Vatican II regarding limbo?
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The origin of limbo is actually the place where the Saints that came before Jesus “waited”. This is what the Apostles' Creed speaks of by calling it the dead:
He descended to the dead.
And the CCC clarifies this and uses the term abode of the dead:
Scripture calls the abode of the dead, to which the dead Christ went down, "hell" - Sheol in Hebrew or Hades in Greek - because those who are there are deprived of the vision of God. Such is the case for all the dead, whether evil or righteous, while they await the Redeemer: which does not mean that their lot is identical, as Jesus shows through the parable of the poor man Lazarus who was received into "Abraham's bosom": "It is precisely these holy souls, who awaited their Saviour in Abraham's bosom, whom Christ the Lord delivered when he descended into hell." Jesus did not descend into hell to deliver the damned, nor to destroy the hell of damnation, but to free the just who had gone before him. (CCC 633)
The hypothesis that was abandoned recently implied that children who died unbaptized would go to that same place.
And to answer your other question: I found no mention of the term Limbo (Latin: Limbus) in any of the official Vatican II documents using Google. (All the documents are there and indexed you can verify) So maybe they just didn't treat the subject.
If you look at this question, you will find references to statements by Benedict XVI regarding Limbo. Specifically he says:
Limbo was never a defined truth of the faith. Personally - and here I am speaking more as a theologian and not as Prefect of the Congregation - I would abandon it since it was only a theological hypothesis. It formed part of a secondary thesis in support of a truth which is absolutely of first significance for the faith, namely, the importance of baptism. …. One should not hesitate to give up the idea of 'limbo' if need be (and it is worth noting that the very theologians who proposed 'limbo' also said that parents could spare the child limbo by desiring its baptism and through prayer); but the concern behind it must not be surrendered. Baptism has never been a side issue for the faith; it is not now, nor will it ever be." - Joseph Cardinal Ratzinger
If Limbo was never a doctrine of the faith, then Vatican II cannot have 'eliminated' it. In short, Limbo was only ever a theory, and is less popular these days than it was.