If you receive a K-1 fiancé visa with which to enter the U.S. and get married, your unmarried children under the age of 21, whether or not they are the biological children of your U.S. citizen fiancé, might be eligible to accompany you to the United States. They will be given “K-2” visas. The term "children" includes not only your natural children, but your adopted children and any children born out of wedlock, if your home country legally recognizes them as yours.
Don't be confused by the fact that already-married applicants for U.S. green cards need to prove that their children fit the definition of “stepchildren,” by showing that the parents' marriage took place before the children turned 18. You, as a fiancé visa applicant, don't need to fit those criteria.
The only reason we even mention it is that immigration officials themselves sometimes get confused about it and try to deny K-2 fiancé visas to children who are over 18 but still under age 21. If this happens to you, understand that the official has gotten the laws mixed up, and suggest that he or she read the immigration regulations at 8 C.F.R. § 214.2(k)(3).
Your children will have to go through the same (or a very similar) application process as you. At the beginning of the application process, all the U.S. citizen sponsor needs to do is to include the children's names in Part 2 of the Petition for Alien Fiancé (Form I-129F). The consular officials should then send extra sets of the required forms for the children to fill out.
If you do not receive these extra forms, contact the consulate. For young children, it's okay for a parent to fill out the form, and even sign it, on their behalf. (Just sign your name, then write Parent of [name of your child].)
The children will have to prove that they are not inadmissible and that they will be financially supported along with the immigrating fiance. Each child will need to show proof of their relationship to the immigrating K-1 parent, such as a birth or adoption certificate. You will also need to gather many of the same documents for the children as you do for the main applicant, such as medical reports, photos, and passports; and to pay individual visa fees.
After you get married and apply for a green card (to "adjust status" to lawful permanent residence), your K-2 children may apply along with you. Each will need to submit (by mail) a separate application for adjustment of status to U.S. Citizenship and Immigration Services (USCIS).
For your planning purposes, note that the children must remain unmarried and under age 21 right up to the day they enter the United States on their K-2 visas. Fortunately, if you alert the immigration authorities to an upcoming 21st birthday, they can usually speed up the processing for you. (Unfortunately, a law you may have heard of called the Child Status Protection Act does not protect children on fiancé visas from the loss of visa rights caused by their turning 21.)
Check your own country's law on removing your children if their other parent is staying behind. If you are planning to bring children to the United States who are not the biological children of your fiancé, it will be up to you to comply with your country's custody requirements. Even if the children are legally in your custody, you may need to get written consent from the other parent for you to take the children out of your country.