An incredibly difficult question to ask about Christianity is what happens to children who die? They can't exactly accept Christ, so do they go to Hell? This question is made especially difficult because the Bible actually doesn't address the issue, at least not directly. There isn't a single verse that specifically says "this is what happens to those who die before they're old enough to accept Jesus". But there are, of course, a lot of theories in the Church, and thankfully they're not as ad hoc as one might expect.
The most common theory in the Church is that there is what's called an "age of accountability". Some people are even so confident in this view that they'll ask the leading question of when is the age of accountability?
I want to make it clear that I don't think there actually is an age of accountability. At least, I don't think we can specifically say "the age of accountability is 7", or something like that. A similar concept exists in the Bible, that being the "age for love" (HCSB phrasing). In Ezekiel 16, God describes Israel as being like a baby girl, neglected by her birth parents, but God looked after her. Then ultimately, her breasts formed, her hair grew, and she reached the "age for love". In the laws of more civilised countries, there is a clear age of consent. Ezekiel 16:7-8, however, doesn't give this age. Neither does anywhere else in the Bible. What it does do is describe the state a woman is in when she's at that age, namely her body is fully developed. This, of course, happens at different times for different people. You can't say the age for love is 18, for example, because some people reach it earlier, others reach it later. Putting an age on it certainly helps in a courtroom, as well as protects those who reach it later, but beyond that, it's not really that useful.
Just as the age for love varies according to the individual, so also does the age of accountability. Nevertheless, while it's not easy to say when the age of accountability is, it is easy to make a near unassailable case for the existence of such an age.
The first hint towards the age of accountability is found in one of the earliest scriptures to be written: Deuteronomy 1. Israel rebelled repeatedly on the trip through the wilderness, even yearning to return to Egypt. This rebellion angered God so much that the entire generation, even Moses, would be prohibited from entering the land, except Caleb and Joshua. But verse 39 says that the children would be allowed to enter, because they "have no knowledge of good and evil". This is significant, because the Bible repeatedly affirms that knowledge is required for any punishment to apply. Romans 5:13, for example, tells us that though sin was in the world before the law, it can't be imputed unless the law exists. Romans 7:7-12 further explains that the law actually produces evil desires in us (i.e. we see what is forbidden to us and seek it).
But does that mean if it's not written down for us, we're safe? No, unfortunately, because while the law gives us specifics, we have inborn consciences which serve as a law for us (Romans 2:14-15), so although a lack of specific knowledge granted by way of the scriptures will result in less punishment (Luke 12:47-48), there is unfortunately a time in every person's life when, educated or not, we lose the excuse of youth.
Just as Deuteronomy 1:39 tells us that the children of Israel were too young for knowledge of evil, and thus were spared the punishment of not entering the land, both Paul and Jesus told us to imitate children. Three times in the Bible, we're given an account of when some children were brought to Jesus, but the apostles tried to shoo them away. Jesus wasn't too happy about this and told them "let them come to me, and do not hinder them, for the Kingdom of Heaven belongs to such as these" (Matthew 19:14; Mark 10:14; Luke 18:16). In Matthew 18, Jesus actually goes one step further. In response to the question "Who, then, is the greatest in the kingdom of Heaven?", Jesus takes a child and uses him/her as an example. He says "Assuredly, I tell you, unless you are converted and become as little children, you will by no means enter the Kingdom of Heaven."
These two concepts are the ones most commonly cited in defence of the age of accountability. If Heaven belongs to "ones such as these", who is more likely to go to Heaven than ones such as these? Jesus sets children forward as the pattern of conduct because they are innocent. They don't know good and evil.
But then we come to Paul, who, like Jesus, implores us to be like children. But not with regard to understanding. In 1 Corinthians 14:20, he says "Brethren, do not be like children in understanding; however, in malice, be babes, but in understanding be mature." Combined with Jesus' teaching, we see that we're not supposed to be children with regard to intellect. Children are spared from sin because they don't know any better (and younger children can't physically do anything that even resembles sin), but we're supposed to avoid it because we do know better. But Paul tells us to be like children with regard to malice.
I want to tell you a story. A man went to an African village and challenged the children to race each other. Whoever won the race would be given an orange. Think about that: You live in a deprived area, and some rich guy comes along with a tasty, foreign fruit. How motivated are you to win that race? But the children had a greater motivation. They didn't run to win, but rather they all kept pace with each other. They all crossed the finish line at exactly the same time, so the prize was split between them. When they were asked why, they responded by asking what's the point in just one of them being happy if the rest aren't? This wonderful story is the very picture of 1 Corinthians 14:20.
But doesn't 1 Corinthians 14:20 say babes, not infants? Isn't it the weakness of a baby's limbs that make them unable to sin? Yes and no. While a baby obviously can't sin because it can't even do much more than poop and cry, we have to remember Jesus' teaching as well. In Matthew 18, Jesus didn't just pick a random baby, He actually called to the child (v2), and said "whoever humbles himself as this little child is the greatest in the Kingdom of Heaven" (v5, emphasis mine). So, clearly the child is mature enough to both understand when they're being summoned, and physically developed enough to approach the one who summoned them. The age of accountability, therefore, clearly extends a little further than the neonatal stage.
It's interesting to note that Jesus Himself seems to have gone through a moral development stage. Of course, Jesus would never have sinned regardless of His upbringing, but in the prophecy of His birth in Isaiah 7, specifically in verses 15-16, it's said that He would eat curds and honey so He would know to refuse evil and choose good. In other words, He goes through the exact same process as every other human being, but whereas our sinful nature causes us to go astray as soon as we are able, Jesus never went astray. But there was a time in His life where He didn't even know. He was a normal baby, with as blank a mind as any other.
So what happens to infants when they die? Sadly, the answer isn't as clear as the brightest summer's day, but it's indisputable that every human being goes through a developmental stage wherein they cannot be judged for sin. The imputation of sin requires knowledge of good and evil, which children simply do not have. Regardless of when the age of accountability may be, this saying rings true: "Teach your children to love Christ, because the world will teach them not to." It's never too early to try to secure your child's salvation.