I get about one-two flats a year. I still carry a repair kit, though, because when I do get a flat, it's usually at a time and place when the only person who can fix said flat is me.

I think big, wide, squishy tires are much less prone to flats than skinnier tires. I've gotten a lot more flats since I switched from 37mm to 28mm tires. I'm seriously considering going back to the 37s.