In many states, they can. Unfortunately, even in states that permit joint adoptions, some social service and government agencies discriminate against unmarried couples. An unmarried couple can expect to do a bit more work to prove that their home is a stable and healthy environment for raising children. It's wise for any unmarried couple to consult a good family lawyer to get some practical guidance before moving ahead with adoption plans.
If an unmarried couple jointly adopts a child, they will both be legal parents. This means that each partner has equal legal responsibilities to raise and support the child. And, if the partners ever separate, each has an equal legal right to petition a court for custody or visitation of the child, as well as an obligation to provide child support.
For more information on adoption, see the Adoption area of Nolo's website.