The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most \frac{1}{2}(3+\sqrt{1+8k}) and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a directed cycle on 5 vertices which have oriented chromatic number 5. This bound is tight for k\le 4.