| Christ died so we may|
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|A multi-chef broth|
|Devil's in the details|
|The pearly gates|
is the party dance popular in the Caribbean was the way-place where unbaptised babies went after death, at least according to Roman Catholic wishful thinking. Hell and even purgatory seemed a bit too harsh a punishment for the crime of dying before their parents could baptize them, so Limbo was invented as a slightly less bad form of eternal isolation. Although it is not part of official Catholic doctrine, it was widely accepted within the church until 2007, at which point the Pope apparently realized that it didn't actually exist.
The doctrine of original sin left Catholics in an awkward situation. Since we are all born with original sin—baptism being the only way to remove it—babies who die prior to being baptised would seem destined to burn in Hell. St. Augustine, nice guy that he was, believed that babies were indeed going to be tortured for all eternity for the heinous crime of being born. This line of thinking remained popular until medieval scholars revived the heretical idea of Limbo.
Limbo gained wide acceptance, although it was never formally accepted as doctrine. Bereaved parents perhaps drew some comfort from the knowledge that their babies were happy in Limbo for eternity. 
Problems with Limbo
Babies had been saved from hell, but Limbo seemed a little unfair. If hell is an eternal prison of torture then Limbo was similar but with natural happiness instead of torture. Unbaptised babies could not go to Heaven without the doctrine of original sin being seriously undermined. Allowing unbaptised babies entry to Heaven would mean that anyone with a good reason for not being baptised (e.g. people living in countries with no Catholic priests) would also get to go to heaven, and that would render the Catholic Church obsolete. So basically, the Church was faced with two choices:
- Invent the concept of Limbo.
- Relinquish all power by saying God treats all religions equally (and we can't have that now, can we?).
Hence Limbo was invented and everyone was happy, except the dead babies who were too busy participating in dead baby jokes to care.
Officially, the Church never recognized the notion of Limbo as a separate "place" from Heaven, Purgatory, and Hell, although they never flat-out denied its existence until 2007. In Dante's Inferno, the author tried to reconcile this by making Limbo into the outermost circle of Hell, devoid of any actual torture (but still a rather gloomy place). Of all the parts of the Inferno, the author rather quickly passes through it, emphasizing in narration that he doesn't have enough time to talk about it.
In 2007, Pope Benedict XVI decided that Limbo doesn't actually exist. There remains no clear answer as to the fate of unbaptised babies, but most Catholics will continue doing what they always do – assume that God wouldn't do anything nasty to the little babies.