The United States government runs the Children's Passport Issuance Alert Program (CPIAP), which may be very valuable to you if you got divorced, and you share custody of your children.
The program gives the Department of State's Office of Children's Issues the power to get in touch with parents who are enrolled in the program to find out if they gave their consent for the other parent to apply for a passport on the child's behalf. You can enroll your child and then trust that you will get that phone call before the government hands out a passport, even if everything else appears to be in order.
If you think that your ex may already have applied for a passport, remember that this process takes time, and it may still be pending. You can enroll, and the government will tell you if the request is pending or if the passport has already been issued for your child.
The main qualifier is that your child has to be a minor, under 18 years old and that he or she has to be a U.S. citizen or qualify as such.
Why would you want to enroll your child in this program? Some divorced parents have very real concerns about abduction, especially in contested cases. The system may provide you with a warning that essentially tells you that your ex is attempting to take the child out of the United States. You can then act to prevent this from happening, protecting your rights and keeping the child safe. It is critical to do this correctly, so make sure you know exactly what steps to take.