  After a brief introduction into the use of Calabi--Yau varieties in string
dualities, and the role of toric geometry in that context, we review the
classification of toric Calabi-Yau hypersurfaces and present some results on
complete intersections. While no proof of the existence of a finite bound on
the Hodge numbers is known, all new data stay inside the familiar range
$h_{11}+h_{12}\le 502$.
