If you are applying for or receive refugee or asylum status in the United States, you are probably concerned with what will happen to your family members, whether they live in the U.S. or overseas—particularly if they were not also persecuted, and do not have a separate claim for asylum or refugee status.
The good news is that, if you are married or have unmarried children under the age of 21, your spouse and children can be granted refugee or asylee status along with you, or after you, as your "derivatives."
To achieve this, they will need to provide proof of their family relationship to you. This typically means birth and/or marriage certificates (with English-language translations if you're applying for asylum, within the United States.) If you don't have a marriage certificate because you entered into an informal or "camp" marriage while fleeing persecution, and thus had no other choice, USCIS can make an exception. (See its 2022 Revised Guidance on Informal ("Camp") Marriages.)
Like you, they will be allowed to live and work in the United States, and to apply for lawful permanent residence within one year of obtaining their status. (See further discussion of application process below.)
As an asylee or refugee you will not, unfortunately, be able to obtain derivative status for more distant relatives, such as parents, brothers, or sisters.
If you would like to help more distant relatives such as these to immigrate to the United States, you will need apply for U.S. citizenship as soon as possible after becoming a lawful permanent resident, as described in these articles about U.S. Citizenship. Then you can petition to have your parents, married children, children over age 21, and siblings immigrate to the United States.
However, other than your parents (who are considered "immediate relatives"), these relatives will likely face years-long waits before a green card (immigrant visa) becomes available to them. That's because of annual limits on the visa numbers given out, and a resulting waiting list based on "priority date."
If you're an asylee whose family is already in the U.S. with you, simply including their names and information on your application for asylum will be enough to get them granted asylee status along with you. And refugees can bring family members along on their application.
That means naming family members on either Form I-590 (for refugees) or Form I-589 (for asylees). (You must name them in any case, whether or not they are currently with you or intend to come to the United States.)
Be sure to also provide evidence of the family relationship, such as copies of your marriage certificate or your children's birth certificates. If you do not have the actual certificates available, you may submit other forms of evidence, such as medical records or affidavits naming you as the person's spouse or parent. See the instructions that come with the form for details.
If your spouse and children experienced separate persecution, then having them submit their own applications for refugee or asylum status is also possible. In fact, it might be a good idea, in case your application is denied. Then you would become your spouse or child's derivative.
If you are traveling separately from your spouse and children when you are granted refugee status, or they are outside the U.S. when you are granted asylum, you will need to submit a separate application asking that they be allowed to join you.
There's a deadline for this. You must submit the application within two years of obtaining your refugee or asylee status.
Your request must be made on Form I-730 (available from USCIS for free download). Submit the form to the USCIS Service Center indicated on the website. There is no fee to submit this form.
Eventually, both your family members and you will be called in for interviews at a USCIS or consular office, depending on where they and you live at the time. With any luck, they will be granted the right to enter the United States.