  During the last years we have generated a large number of data related to
Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive
polyhedra. We classified all reflexive polyhedra in three dimensions leading to
K3 hypersurfaces and have nearly completed the four dimensional case relevant
to Calabi-Yau threefolds. In addition, we have analysed for many of the
resulting spaces whether they allow fibration structures of the types that are
relevant in the context of superstring dualities. In this survey we want to
give background information both on how we obtained these data, which can be
found at our web site, and on how they may be used. We give a complete
exposition of our classification algorithm at a mathematical (rather than
algorithmic) level. We also describe how fibration structures manifest
themselves in terms of toric diagrams and how we managed to find the respective
data. Both for our classification scheme and for simple descriptions of
fibration structures the concept of weight systems plays an important role.
