How did the second Vatican council (aka Vatican II) address the concept of limbo? 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?
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.