Islamiclly how do we punish young children committing criminal acts, e.g. killing someone? I asked this question, because a lot of young children, who are not yet mature are committing big crimes

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The punishment of children who have not yet reached the age of maturity is to inflict on them what would be enough to make them fear committing crime and sins. This is defined by the ruler.

Retributive punishment (Hadd) is not to be inflicted on them as long as they have not reached the age of maturity. The evidence for this is the Hadeeth in which the Prophet said: “The pen is raised for three (meaning: there is no obligation upon three): one who is sleeping until he wakens, the child until he becomes an adult, and one who is insane until he becomes sane.”

Imam Maalik said: “The consensus is established on the fact that there is no Qisaas (retributive punishment) on the children and that killing them is a mistake as long as they have not reached maturity or the Hudud (plural of Hadd) have not become due on them.”

So, if the child gets matured, or if his pubic hair grows, or if he reaches fifteen, then he will be treated as any other mature person provided he is sane.

Allah knows best.