  This article is about chromatic numbers of hyperbolic surfaces. For a metric
space, the $d$-chromatic number is the minimum number of colors needed to color
the points of the space so that any two points at distance $d$ are of a
different color. We prove upper bounds on the $d$-chromatic number of any
hyperbolic surface which only depend on $d$. In another direction, we
investigate chromatic numbers of closed genus $g$ surfaces and find upper
bounds that only depend on $g$ (and not on $d$). For both problems, we
construct families of examples that show that our bounds are meaningful.
