My father, a French national, wants to give me (US tax-paying citizen) money. Does the $13000 limit apply? If so, can he give to my wife and 2 kids up to the same limits? Do the transactions have to be separated? As recipients, do we need to fill out any paperwork?

There are NO tax ramifications for you if your father (who is NOT a U.S. citizen) chooses to give you any amoney in ANY amount.

U.S. gift tax laws laws apply ONLY to U.S. residents and only to the person who is GIVING the gift, NOT to the recipient.

You MAY need to complete Form 3520 at the bank receiving the wire transfer to document from where the money came, but no taxes will be due.

As ATE said, U.S. gift taxes apply to U.S. citizens and U.S. domiciled residents. However, U.S. gift taxes also apply to nonresident aliens for tangible property situated in the U.S. Note that the IRS argues that cash and checks are tangible property and a gift doesn't take place until the recipient accepts it.

A wire transfer from a foreign bank account of a nonresident alien to a foreign or U.S. bank account of a U.S. citizen/resident would likely not be subject to U.S. gift tax.

The $13,000 annual exclusion applies for nonresident aliens as well. Therefore, even if the gift is of tangible property situated in the U.S. gifts of $13,000 to each individual can be made without triggering gift tax.

If a nonresident alien does make taxable gifts in excess of $13,000 in a year, gift tax would be due for that year (i.e. the $1,000,000 lifetime exclusion is not available for nonresident aliens).