Let E be a compact set with connected complement in the complex plane, and let A(E) be the class of all complex continuous function on E that are analytic in the interior of E. Let f be a function from A(E), which is zero free on the interior of E. By Mergelyan’s theorem f can be uniformly approximated on E by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on E? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets E. The talk describes classes of functions for which zero free approximation is possible on an arbitrary E.