Can Barbarians take over a city?

I was playing the other day and saw barbarians actually attack a City State. They got its walls down to about 50% but couldn't take the city. I think it was a CS anyway... my memory isn't great. :)

I haven't seen this before. It made me wonder: What happens if the barbarians take a city? Or a CS? I remember in previous Civ games barbarians could take cities, and sometimes did, but I've never seen it in Civ 5.