  According to a recently proposed scheme for the classification of reflexive
polyhedra, weight systems of a certain type play a prominent role. These weight
systems are classified for the cases $n=3$ and $n=4$, corresponding to toric
varieties with K3 and Calabi--Yau hypersurfaces, respectively. For $n=3$ we
find the well known 95 weight systems corresponding to weighted $\IP^3$'s that
allow transverse polynomials, whereas for $n=4$ there are 184026 weight
systems, including the 7555 weight systems for weighted $\IP^4$'s. It is proven
(without computer) that the Newton polyhedra corresponding to all of these
weight systems are reflexive.
