Sheikh `Abd al-Wahhâb al-Turayrî, former professor at al-Imâm University in Riyadh
Adoption of children for the purpose of bringing them up and caring for them is permissible. It is in fact a good deed to adopt and bring up children in need, like orphans and foundlings. There is great reward in doing so.
At the same time, it is unlawful to attribute the child to the man who is bringing him up. Allah says: “Call them by the names of their fathers; that is more just with Allah. But if you know not their father’s names, (then they are) your brothers in faith, or your wards.” [Sûrah al-Ahzâb: 5]
It is lawful to bring up children in any house, but they must always be attributed to their true fathers. The child will carry his real father's name. Likewise, the child will not inherit from his guardian who adopted him and the child may even marry from his adoptive “father’s” children.
Source: Islam Today