As this has come up again, Joe Heschmeyer has an excellent description of the current state of things:
So what happens to a baby who dies without Baptism? After all, neither Heaven or Hell seem like an obvious fit. On the one hand, they weren’t baptized, so they seemingly can’t enjoy the Beatific Vision: “For whoever would draw near to God must believe that he exists and that he rewards those who seek him” (Hebrews 11:6). On the other hand, they didn’t sin, so it would be unjust to damn them.
For centuries, Catholics tended to believe that they went to Limbo. On surface, this idea doesn’t look Biblical: you won’t find the word “Limbo” anywhere in Scripture. What you will find, though, is the idea that the faithful who died before Christ descended into what’s “the Bosom of Abraham.” This is where Jesus describes Lazarus as going after death (Luke 16:23). It’s hell, inasmuch as it’s not the perfect rest and enjoyment of the Beatific Vision of Heaven… but it’s not damnation, and there’s no sense that Lazarus or anyone else in Abraham’s bosom suffered.