Assume that an function analytic f in the unit disk \(\mathbb {D}\) of the complex plane has Taylor coefficients belonging to a finite set. If we also assume that f belongs to the Nevanlinna class. or even to Hardy class \(H^p\left (\mathbb {D}\right )\) , \(p\in \left (0,1\right )\) , does it imply that f is a rational function? It is worth noting that if assume that the Taylor coefficients are integers (instead of being taken from a finite set), this problem is solved completely. The article contains a review of results starting from work of P. Fatou, F. Carlson and G. Szegö, open problems, and some new results hoping to stimulate interest to this old but very exciting subject.