A marriage must be legally valid and existing, and must not have been entered into for the sole purpose of obtaining permanent residence in the US. In other words, it must not be a fraudulent or "sham & marriage. All prior divorces by either spouse must also be legally valid. The USCIS will recognize a foreign marriage as long at it conforms to the formal requirements of the jurisdiction it was entered into.
Polygamous marriages are not valid for immigration purposes, even if they were valid in countries or states where they were solemnized. Same sex marriages are now recognized by the USCIS. The process for Legal permanent residence based on marriage to a US.citizen is initiated by filing a petition for Alien Relative (I-130) with the USCIS, along with an adjustment of status application (I-485). The final step in the process is the attendance by the couple at an interview with a USCIS officer who will determine, based upon the documentary and verbal evidence presented by the couple, whether or not the applicant is eligible for permanent residence. The beneficiary must submit a medical report, and the petitioner (sponsor) must demonstrate the financial ability to sponsor the beneficiary based on the USCIS Affidavit of Support minimum poverty guidelines.
Upon the successful conclusion of this interview, the applicant with be issued with either conditional permanent resident status, when the marriage is less than two years duration,or permanent residence when the marriage is over two years.
A conditional permanent resident must reapply for permanent residence within 90 days of the expiration of the two year period of his/her conditional residence.