We show that the maximum number of strictly star-shaped polygons through a given set of n points in the plane is (n 4). Our proof is constructive, i.e. we supply a construction which yields the stated number of polygons. We further present lower and upper bounds for the case of unrestricted star-shaped polygons. Extending the subject into three dimensions, we give a tight bound of (n 9) on the number of distinct sets of star-shaped polyhedra. 