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According to a recently proposed scheme for the classification of reflexivepolyhedra, weight systems of a certain type play a prominent role. These weightsystems are classified for the cases $n=3$ and $n=4$, corresponding to toricvarieties with K3 and Calabi--Yau hypersurfaces, respectively. For $n=3$ wefind the well known 95 weight systems corresponding to weighted $\IP^3$'s thatallow transverse polynomials, whereas for $n=4$ there are 184026 weightsystems, including the 7555 weight systems for weighted $\IP^4$'s. It is proven(without computer) that the Newton polyhedra corresponding to all of theseweight systems are reflexive.Comment: Latex, 14 page

WEIGHT SYSTEMS FOR TORIC CALABI-YAU VARIETIES AND REFLEXIVITY OF NEWTON POLYHEDRA