Polygon-circle graphs are intersection graphs of polygons inscribed in a circle. This class of graphs includes circle graphs (intersection graphs of chords of a circle), circular arc graphs (intersection graphs of arcs on a circle), chordal graphs and outerplanar graphs. We investigate binding functions for chromatic number and clique covering number of polygon-circle graphs in terms of their clique and independence numbers. The bound α log α for the clique covering number is asymptotically best possible. For chromatic number, the upper bound we prove is of order 2^ω, which is better than previously known upper bounds for circle graphs.