Additional rights will become yours with time, such as that to apply for a U.S. green card after one year, and to apply for U.S. citizenship four years after that. Learn more about how to protect and make the best use of your refugee or asylum status here. However, you MUST apply for the Green Card after the one year grant. It is not automatic and will not happen unless you apply.

Question: Can I bring my spouse and children into the U.S. now?

Answer: Once you have been granted asylum, your immediate family members (spouse and children)—whether they are in the U.S. or outside—are entitled to a “derivative” grant of asylum. If your spouse and children were included in your asylum application and are physically present in the U.S., they will automatically receive asylum at the same time as you.
If they are overseas, or were not included in your application, you can file USCIS Form I-730, Refugee/Asylee Relative Petition to obtain asylum for them. Use a separate form for each family member.

For your spouse to be eligible for asylum, the two of you must have been legally married (that is, with a government-issued certificate) before you were granted asylum. For your children to be eligible, they must be unmarried and younger than 21. Thereafter, once you qualify for the Green Card or residency, they will as well.