A $1$-planar graph can be drawn in the plane so that each arc is crossed at most once by another arc. A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times.

Planar graphs are $4$-colorable, and $1$-planar graphs are $6$-colorable.
($K_6$ is $1$-planar: image below from here.)

What bounds are known on the colorability of $k$-planar graphs?

This has likely been studied; I just don't know the results. Thanks for pointers!

**Answered**by Tony Huynh and David Eppstein: The chromatic number of a $k$-planar graph is $\Theta(\sqrt{k})$.