Who gets custody of young children after a divorce in Islamic law?

In the majority of classical fiqh, the mother has the first right to custody (hizanah) of young children after divorce, on the reasoning that a young child's welfare is best served by the mother's direct care during the early years. In the Hanafi school (the most widely followed in South Asia), this default lasts until a son turns seven and a daughter reaches puberty, after which custody is expected to transfer to the father; the Maliki school extends the mother's custody of sons until puberty and daughters until marriage, while the Shafi'i and Hanbali schools generally give the child a degree of choice between parents once they reach an age of discernment (roughly seven to nine).

Across all schools, the father remains financially responsible for the child's maintenance regardless of who has physical custody, and a mother's right to custody is commonly understood (in most, though not all, schools) to be affected if she remarries someone who is not a close relative (mahram) of the child — the reasoning being the child's comfort and welfare in that household, not a penalty against the mother. It's also important for readers in India to know that Indian family courts, when a custody dispute is actually litigated, apply the "welfare of the minor" standard under the Guardians and Wards Act, 1890, as the paramount consideration — this can override the default fiqh age-based rule if the court finds it genuinely serves the child's best interest to do otherwise.

Source: general Sunni fiqh on hizanah, with the age-based Hanafi position most commonly followed in South Asia; Guardians and Wards Act, 1890 (India) on welfare-of-the-minor as the governing civil standard.