What are the differences between the K-1 visa and the K-3 visa?

These visas are very similar. The appropriate one to seek depends on whether or not the couple is married.  The K-1 visa applies to fiances (unmarried only), and the K-3 visa is used if the couple is already married.  Once the marriage occurs, the beneficiary is no longer eligible for the K-1 but is eligible for the K-3 visa.  In order for a spouse to qualify for the K-3 visa, she/he must also be the beneficiary of an I-130 immigration visa petition filed by the US citizen spouse.