Question: Will disbelievers' children who die before reaching the age of puberty and discretion enter Paradise or not?
ANSWER
Islamic scholars put forth seven different deductions on this issue. They are as follows:

1. Disbelievers' children who die before reaching the age of puberty and discretion will enter Paradise. As a matter of fact, it is declared in hadith-i sharifs:
(In Paradise, there will be children around Ibrahim 'alaihis-salam. Among them will also be the children of polytheists who died when they were young.) [Bukhari]

(I asked my Lord to forgive the children of polytheists. He granted my wish and admitted them to Paradise.) [Abu Nu'aym]

(Every child is born upon the fitrah of Islam [in a nature well fitted for Islam].) [Bukhari]

2. The children of disbelievers will be servants of Believers in Paradise. It is declared in hadith-i sharifs:
(I asked my Lord that the children of polytheists who died when they were young be at Believers' service in Paradise, and He accepted it.) [Hakim-i Tirmidhi]

(The children of polytheists are servants of the inhabitants of Paradise.) [Tabarani]

3. They will follow their parents. If either parent enters Paradise, the child will enter Paradise as well. If both parents enter Hell, the child will enter Hell, too. When the fate of disbelievers' children who died when they were young was asked to our Master the Messenger of Allah, he said, "They will follow their fathers." When he was again asked how they would follow their fathers without having done any deeds, he replied as follows:
(Allah certainly knows what deeds they would have done if they had grown up.) [Abu Dawud]

Two hadith-i sharifs say (what means):
(Both polytheists and their children will be in Hell.) [Imam-i Ahmad]

(The one who buries a female child alive and the one who is buried will be in Hell.) [Abu Dawud]

[That sinless children will go to Hell is only one deduction out of seven deductions. Those who label the above-mentioned hadith-i sharifs as fabricated with their own personal thoughts are ignoramuses who do not know what hadiths and scholarly deductions are. Different madhhabs emerged because of different hadith-i sharifs. For example, while it is fard in the Shafi'i Madhhab to recite the Fatiha behind the imam, it is makruh close to haram in the Hanafi Madhhab. The reason why these two rulings are so opposite to each other is that our Master the Messenger of Allah stated them differently. Our Master the Prophet stated them differently so that they could be a mercy for his Ummah. Hence, when one has difficulty in doing something according to one's own madhhab, one can do it according to another true madhhab, thus benefiting from the mercy that exists in a different scholarly deduction.]

4. The children of disbelievers would have been either Believers or disbelievers if they had grown up. Since it is known in divine knowledge, the decree will be according to it. That is, those who would have been disbelievers will go to Hell, while those who would have been Believers will go to Paradise. It is stated in a hadith-i sharif:
(Allah certainly knows what deeds they would have done if they had reached the age of puberty and discretion.) [Bukhari, Muslim, Nasai]

5. They will enter neither Paradise nor Hell. They will stay at a place called A'raf, but it will not be permanent because in the Hereafter there will be no place other than Hell and Paradise. Also, there were scholars who said, "They will be tested in the next world. Those who get it through will go to Paradise, while those who fail it will go to Hell."

6. The children of disbelievers will be turned into dust. Having faith (iman) is a precondition for entering Paradise, while being faithless is a precondition for abiding in Hell eternally. The children of disbelievers are neither with it nor without it. Therefore, they will be annihilated. People who live in the mountains, forests, caves, or desert and who have never heard of any religion will not enter Paradise as they do not have faith. They will not enter Hell as they have not heard of Allah, Paradise, and Hell and have not rejected them. They will be annihilated like animals.

7. Some scholars, when faced with different hadith-i sharifs over the matter, remained silent and refrained from saying anything. [These pieces of information have been quoted from reliable works, such as Maktubat-i Rabbani, Faraid-ul fawaid, and Taj.]

The child of a disbeliever and faith
Question: What should be done if the child of a disbeliever becomes a Believer and dies before reaching the age of puberty?
ANSWER
He or she is treated as a Muslim's child is treated.