Arthur Benjamin is a research mathematician (I've actually mentioned him before, although I didn't realize that until I looked at his web page and saw that the title of one his papers looked familiar...) and also a "mathemagician" -- he has a stage show in which he does mental calculations. See this 15-minute video of his show at ted.com.
He starts out by squaring some two-digit numbers... this didn't impress me much, because I could almost keep up with him. (And 37 squared is especially easy for me. One of my favorite coffeehouses in Cambridge was the 1369 Coffee House, and at some point I noticed that that was 37 squared. So I'll always remember that one.) Squaring two-digit numbers is just a feat of memory. Three- and four-digit numbers, though... that's a bit more impressive. And of course I'm harder to impress in this area than the average person.
One trick that might not be obvious how it works: he asks four people to each find a number 8649x (the 8649 was 93 squared, from the number-squaring part of the show) for some three-digit integer x, and give him six of the seven digits in any order; he says which digit is left out. How does this work? 8649 is divisible by 9. So the sum of the digits of 8649x must be divisible by 9. So, for example, say he gets handed 2, 2, 2, 7, 9, 3; these add up to 25, so the missing digit must be 2, to make 27? (How could he tell apart a missing zero and a missing nine? I suspect there's a workaround but I don't know what it is; the number 93 was given by someone in the audience, so I don't think it's just memory.)
He also asks people for the year, month, and day which they were born and gives the date; I found myself trying to play along but I can't do the Doomsday algorithm quite that fast... and I suspect he uses something similar. (I noticed that he asked three separate questions: first the year, then the month, then the day. This gives some extra time. I know this trick well; when a student asks a question I haven't previously thought about, I repeat it. I suspect I'm not the only one.)
The impressive part, for me, is not the ability to do mental arithmetic -- I suspect most mathematicians could, if they practiced -- but the ability to keep up an engaging stage show at the same time.
(The video is on ted.com, which shows talks from an annual conference entitled "Technology, Entertainment, and Design"; there look to be quite a few other interesting videos on there as well.