Why do paintballs curve?

Answer 1

For the most part, Paintballs are without rifling and thus cannot produce a "stablizing" spin as it leaves the barrel, it is stuck with the motion produced by the air flowing behind the ball as it leaves the barrel.

Also Paintballs have a liquid center as opposed to the solid core of a bullet, the liquid inside the ball has a tendency to settle in one side of the ball for than another. This throws the center-weight of the ball off enough to cause a spin of its own.

There are also outside forces that can cause a spin on the ball such as oil or dirt in the barrel, or even the shape of the barrel itself.