On curves of genus g ≥ 2, points differ from each other. For example, every nontrivial automorphism of such a curve has finitely many fixed points, and the set of points that are fixed by some nontrivial automorphism is also finite. In this chapter, we discuss another method of singling out points with special properties. This method, due to Weierstrass, also singles out a finite set of special points on every curve. This set is closely related to the set of fixed points of nontrivial automorphisms, but does not in general coincide with it. In particular, Weierstrass points on a curve do exist even if it has no nontrivial automorphisms.