A graph G=(X, R) defined on a Polish space X is Borel if the binary relation R is a Borel subset of the cartesian product of X with itself.

The Borel chromatic number of such a graph is the least cardinal k such that there is a Borel measurable function c from X to k coloring of the graph, that is, connected elements of X get different images under c.

The study of Borel chromatic numbers was initiated by Kechris, Solecki and Todorcevic (Advances in Mathematics 141 (1999) 1-44) and has received considerable attention since then.

We will survey some of the basic results regarding this concept and mention some open questions.