Answer
It is permissible for a person to adopt a child and taking care of a deserving child on its own earns is a rewarding deed, but the child legally (Islamic Shariah) is not considered his son or daughter; i.e. the child remains non-mahram to him or his wife and he or she does not inherit from the ones who has adopted him or her.
Adoption in the sense to change a child's surname and register him in the official documents as the parents own child is not permissible.