From tsuwm's wwftd:

Quinion notwithstanding, (http://www.quinion.com/words/qa/qa-beg1.htm) I never heard begs the question used in a way that I would have interpreted as meaning raises the question. I've always interpreted the much maligned usage as meaning avoids the question. The idea that the question should be asked is merely a natural consequence of its being ignored, given the context of the situation in which it has been ignored. This usage (avoids the question) is well within the accepted definition of beg. See def 3a in AHD http://www.bartleby.com/61/4/B0160400.html.

I do like his comment that the person who translated [petitio principii] made a hash of it.

well, here's the usage that triggered this for me (again):

The facts continued to beg the question: Did the long arm of Adolf Hitler reach across the sea and into this noisy little American beach town? Had the Nazis hounded this troubled American of German blood until he'd finally put a bullet through his [own] head? "You'll have to ask Hitler when you see him," said the sheriff. "As far as I'm concerned, it's an open-and-shut case."

(from _Two O'clock Eastern Wartime_, a period thriller by John Dunning)

no avoiding the question here, just a device for raising it. it sounded paritcularly false in this WWII setting. but this is the kind of usage I hear or read frequently.

Boy, howdy. That'd rankle my geezer if I heard it used that way. Good thing I ain't never.

I'm glad to see this question brought up because I have always wondered a little at the correct use of it. I have had this fuzzy notion that it meant not only avoiding the question but dodging it using some sort of faulty logic. e.g. "Why did the chicken cross the road?...[you know the rest]"

There are a number of sites dealing with ‘begging the question’, some probably better than this, but this was the most amusing (the only one even slightly amusing?):

http://www.roomours.co.uk/ryder3.htm