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When a person, or couple, legally adopts a child, they typically become the parents of the child for all legal purposes, with absolutely no legal distinction between biological and adopted children.
In many states, even those which do not recognize same-sex marriage, same-sex couples are allowed to adopt children, with both partners being treated as the legal parents.
The state cannot issue and enforce licenses to breed (as much as some might like to). However, state adoption agencies can decide who gets to adopt a child. Obviously, it is preferable for a child to be brought up by a stable, loving family, rather than being a ward of the state, but states will not allow just anyone to adopt.
Generally, anyone wishing to adopt a child has to show that they are financially able to support the child, that they are psychologically and emotionally stable, and that they can provide the child with a safe living environment.
A state authority might also consider the age and physical health of the adoptive parent(s), to ensure (to the extent that one can be certain about such things) that they will be able to care for the child until he or she is independent.
There are many reasons why an individual or couple might want to adopt a child. The most obvious one is that a couple is unable to have children, for whatever reason. Or, the parents might believe that the world is already overpopulated, and it's unethical to bring another child into the world when there are so many who lack decent homes.
Also, if a person gets married, and the spouse already has children from another relationship, the step-parent might want to adopt the spouse's children, and become their other legal parent. This is especially appropriate when the child is very young and/or the other biological parent is not in the picture.
When petitioning to adopt a child, anyone else who has a vested interest in the child's well being is given an opportunity to object, or otherwise make their views known. These include the child's biological parents, current legal guardians, the adoption agency, and (if the child is above a certain age), the child himself.
Legal Effects of Adoption
Once an adoption is finalized, the legal effects are profound: the legal relationship between the adopted child and the parents is identical to the legal relationship between parents and their biological child.
This means that if a parent dies without a will, the child will be entitled to a portion of the parent's estate. The adoptive parents are also entitled to make all of the same decisions that they would be able to make for a biological child, such as what schools to send them to, whether to file lawsuits on the child's behalf, among many others.