This paper considers compact triangulated polyhedra of genus zero. Let V n = {v 0 , v 1 ,. .. , v n+2 } be a fixed labeled collection of n + 3 vertex points on the unit sphere S 2 and denote by T n = {T (1) n , T (2) n ,. . .} the set of all triangulations of S 2 whose vertices are the points of V n. For a given triangulation T (i) n let C (i) n = (d(v 0),. .. , d(v n+2)) denote the degree sequence of the triangulation (something the author calls the combinatorics of the triangulation). Two degree sequences are called equivalent if they differ by a permutation of the vertices of V n. This paper presents an algorithm, or procedure, to list all the distinct triangulations, equivalent or not, for a given point set V n. Reviewer: Geir Agnarsson (Fairfax) MSC: 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)