Limbo can refer to two distinct places.
“Limbo of the Fathers” - the place where the righteous dead went before Christ died and opened the gates of heaven for them. This is in the Bible - see 1 Peter 3:19. This Limbo surely exists.
“Limbo of infants” - the place where unbaptized babies go. This isn’t explicitly mentioned in Scripture (but neither is the Trinity) but can be inferred from the doctrine that all must be baptized to go to heaven (see John 3:3).
This second Limbo is
still a valid theological opinion. The Church has never formally defined it not formally repudiated it. The closest she has ever gotten was Pius VI in
Auctorem Fidei where he called out the Jansenists for teaching children to out and out deny the existence of limbo (Pius uses the word). The Jansenists were teaching Augustine’s theory that the unbaptized were in Hell. Note, Pius doesn’t declare limbo exists, he simply rebukes the Jansenists for teaching that limbo doesn’t exist, Pius is leaving the door open.
Some of the greatest minds in Church History have taught it, including St. Thomas Aquinas the greatest doctor in the history of the Church (
see HERE), and various Church Fathers including St. Gregory Nazianzen , Tertullian, and St. Ambrose. St. Augustine didn’t teach limbo, instead he taught that unbaptized infants would go to Hell, but have the mildest of all punishments.
More recently, in the twentieth century (even before V2), theologians began contemplating the idea that the unbaptized go to heaven. The Church has never officially endorsed this position. The Congregation for the Doctrine of the Faith (then called simply the Holy Office) issued a statement in 1958 urging parents to baptize their children ASAP, noting that the state of the unbaptized after death is unknown,
“The practice has arisen in some places of delaying the conferring of Baptism for so-called reasons of convenience or of a liturgical nature” a practice favored by some opinions, lacking solid foundation, concerning the eternal salvation of infants who die without Baptism. Therefore this Supreme Congregation, with the approval of the Holy Father, warns the faithful that infants are to be baptized as soon as possible…"
The Catechism of the Catholic Church likewise leaves the door open stating, in paragraph 1261:
“As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus” tenderness toward children which caused him to say, “Let the children come to me, do not hinder them,” allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church’s call not to prevent little children coming to Christ through the gift of holy Baptism"
Most Catholics today think Limbo has been rejected in favor of the idea that all unbaptized babies are saved (which dovetails with an increasing tendency toward universalism in the minds of many Catholics). This simply isn’t the teaching of the Church. The fact is, we really don’t know what will happen to them and Catholics are free to decide for themselves between Augustine’s theory (they go to hell), Aquinas’ (they go to limbo), and simply trusting in God’s mercy (they go to heaven) - but Catholics are not free to teach any of these as absolutely certain.
Hope that helps.