Sorry, I should have specified felon rather than simply having a criminal record. Generally speaking any crime that carries a possible prison sentence over a year, though you're right that it can vary by state etc. And blackpowder guns and a few others aren't legally considered "firearms" by the ATF, so a felon can own them.
Which is why in The Dukes of Hazzard, Bo & Luke carried compound bows (with dynamite tied to the arrows!). Because they were moonshiners (a federal felony) they couldn't have guns.
