Can you marry someone who is in the United States on a temporary visa and how can you keep them here?

If their Visa hasn't expired yet it is best for them to go back to their country. THEN you can apply for a K-1 fiancee visa and they can come back to get married within 3 months. Then you apply for a permanent resident and then citizenship. Trust me, it is A LOT EASIER is you don't break the law by letting the visa expire.