My 3-y/o will fall asleep during the day if I can get him still long enough (car rides, "come snuggle with me"), but if he stays active, he will usually stay awake. I think he still needs a nap sometimes even when he doesn't take one (gets pretty crabby, or falls asleep at the supper table), but he's been *not* napping more than he has been napping at least since he was two and a half, and probably more. I think he ought to take a nap every other day or every third day, but sometimes that doesn't happen. However, he usually sleeps extra at night to make up for it.
My older son (almost 5) will occasionally fall asleep and/or need a nap still. He was probably nearly four and a half before he stopped taking regular naps. That was actually the problem with my younger son stopping his naps -- I'd put them both down for a nap at the same time (they share a room), and they'd take a nap; then my older son got to where he didn't *need* a nap, and they started playing too much instead of sleeping, which got my younger son used to the idea of no naps. They're close enough in age that they do most stuff together anyway, so it has been very difficult for me to get my younger son to take a nap without his older brother in bed with him.
My sister's youngest son gave up naps around the time he turned two. But he normally would sleep 13 hours or more at night, so he didn't necessarily *need* a nap either. All kids are different!