I believe what I was taught as a child - that they go to Limbo (which is not a bad place, in fact it is just like heaven except God is not there.) Then on Judgement Day they will be judged just like everyone else and would most likely go to Heaven since they didn't live long enough to sin.
What do YOU think happens to babies who die before the age of reason?
Same thing that happens to anyone who dies after the age of reason - whatever that is. My best guess is that there is a God, and a continuation of our consciousness.
The idea of an all knowing, all loving God who creates living, feeling beings - knowing full well most of them will be tortured forever in Hell - is nonsense, IMO.