Is there a structure theorem for such varieties?
If X is a smooth and proper/projective variety whose canonical bundle $\omega_X$ has finite order in the Picard group, do we know anything about X?
EDIT: As was pointed out in the comments, if $\omega_X^n = O_X$ then one can find a cycling covering of order n, $Y \to X$, with $\omega_Y = O_Y$. So the problem reduces to understanding varieties with trivial canonical bundle. I'll leave the question as is for a few days more, in case someone else wants to contribute. If nothing happens I will delete it.