Do I owe gift tax when I give money to my kids, or do they?

Neither of you owes tax in the vast majority of cases. The giver is responsible for any gift tax, never the recipient, and gifts up to the annual exclusion (an IRS amount adjusted yearly) require no filing at all. If you exceed the annual exclusion to one person, you file Form 709, but you still won't pay tax; the excess simply reduces your large lifetime gift and estate exemption, which most people never come close to using. Your children owe no income tax on money you gift them. The main exceptions are that appreciated assets you gift carry your original cost basis, so the recipient inherits your built-in gain, and gifts to a non-citizen spouse have their own separate limit.